2588:
2561:
867:"It does not show how the data are exhibited by canceling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i. e., "throwing of the problem into an equation" and the transformation of the premises, nor with the subsequent steps, i. e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form."
2530:
2495:
2576:
2623:
2611:
176:
2364:
2546:
35:
228:
2515:
1262:
1194:
curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs.
1128:
960:
1140:
3540:
2479:
518:
330:
27:
2587:
2743:
describe it this way: "The trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error . . . An inference is the dropping of a true premiss ; it is the dissolution of an implication" (p. 9). Further discussion of this appears in "Primitive
988:
curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color. When the number of sets grows beyond 3 a Venn
585:
102–105). The weak point about these consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit
1193:
Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of
509:"... of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose–somewhat at random, as they happened to be most accessible–it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the
2384:
3219:
2379:
545:“In fact ... those diagrams not only do not fit in with the ordinary scheme of propositions which they are employed to illustrate, but do not seem to have any recognized scheme of propositions to which they could be consistently affiliated.”
1305:; from this table the Venn and/or the Karnaugh map are readily produced. By use of the adjacency of "1"s in the Karnaugh map (indicated by the grey ovals around terms 0 and 1 and around terms 2 and 6) one can "reduce" the example's
2307:
connective being the implication that results in the tautology). For example, in the truth table, on the right side of the implication (→, the major connective symbol) the bold-face column under the sub-major connective symbol "
2381:
127:
that each depict a set or category. How or whether these shapes overlap demonstrates the relationships between the sets. Each curve divides the plane into two regions or "zones": the interior, which symbolically represents the
2529:
2494:
223:) The small text to the left erroneously says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise", a book which was actually written by Johann Christian Lange.
2744:
Ideas and
Propositions" as the first of their "primitive propositions" (axioms): *1.1 Anything implied by a true elementary proposition is true" (p. 94). In a footnote the authors refer the reader back to Russell's 1903
166:
of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color.
917:"The Karnaugh map is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram."
1309:
i.e. (x'y'z' + x'y'z) + (x'yz' + xyz') to just two terms: x'y' + yz'. But the means for deducing the notion that "No X is Z", and just how the reduction relates to this deduction, is not forthcoming from this
905:"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6."
2765:
need not be a tautology (a so-called "tautological implication"). Even "simple" implication (connective or adjunctive) work, but only for those rows of the truth table that evaluate as true, cf
Reichenbach
2610:
899:. For example, Hill & Peterson (1968) present the Venn diagram with shading and all. They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement:
2560:
989:
diagram becomes visually complex, especially compared to the corresponding Euler diagram. The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets:
2380:
30:
Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is a disjoint set (it has no members in common) with "animals"
3459:
586:
in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions. ... This defect must have been noticed from the first
2575:
1036:
923:
The history of
Karnaugh's development of his "chart" or "map" method is obscure. The chain of citations becomes an academic game of "credit, credit; ¿who's got the credit?":
337:, but the Veitch is not particularly useful for reduction of formulas. Observe the strong resemblance between the Venn and Karnaugh diagrams; the colors and the variables
2374:
1116:
1076:
2373:
2463:
2367:
3211:
3263:
588:
in the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well
2709:
himself also makes such observations: Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations.
742:
Nowhere is it discussed or labeled, but
Couturat corrects this in his drawing. The correct equation must include this unshaded area shown in boldface:
525:, pp. 115–116 showing his example of how to convert a syllogism of three parts into his type of diagram; Venn calls the circles "Eulerian circles"
1476:
The Truth Table demonstrates that the formula ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) is a tautology as shown by all 1s in yellow column.
1235:~x & ~y & z (From Boolean algebra: 0⋅0 = 0, 0⋅1 = 1⋅0 = 0, 1⋅1 = 1, where "⋅" is shown for clarity)
116:
movement of the 1960s. Since then, they have also been adopted by other curriculum fields such as reading as well as organizations and businesses.
2603:
Some of the Euler diagrams are not typical; some are even equivalent to Venn diagrams. Areas are shaded to indicate that they contain no elements.
2426:
2843:
2371:
629:"Venn's method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only
537:
Venn ends his chapter with the observation illustrated in the examples below—that their use is based on practice and intuition, not on a strict
2116:(by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called the
1301:". Once the propositions are reduced to symbols and a propositional formula ( ~(y & z) & (x → y) ), one can construct the formula's
2545:
231:
The diagram to the right is from
Couturat in which he labels the 8 regions of the Venn diagram. The modern name for the "regions" is
3466:
2456:
2415:
1231:& (logical AND) between propositions; in the minterms AND is omitted in a manner similar to arithmetic multiplication: e.g. x'y'z =
2622:
2383:
3559:
2514:
529:
But nevertheless, he contended, "the inapplicability of this scheme for the purposes of a really general logic" and then noted that,
594:
Whatever the case, armed with these observations and criticisms, Venn then demonstrates how he derived what has become known as his
98:. Unlike Venn diagrams, which show all possible relations between different sets, Euler diagrams show only relevant relationships.
984:
are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2 logically possible zones of overlap between its
2422:
2369:
839:
In modern use, the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the
158:
are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2 logically possible zones of overlap between its
2505:
2449:
549:
Finally, in his Venn gets to a crucial criticism (italicized in the quote below); observe in
Hamilton's illustration that the
365:(1642–1708); however the latter book was actually written by Johann Christian Lange, rather than Weise. He references Euler's
3425:
2501:
3197:
Elementary
Lessons in Logic: Deductive and Inductive. With Copious Questions and Examples, and a Vocabulary of Logical Terms
2377:
2376:
850:(formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "
3509:
2429:
2416:
1228:: 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 1)
947:, Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) a
3295:
3157:
The
Algebra of Logic: Authorized English Translation by Lydia Gillingham Robinson with a Preface by Philip E. B. Jourdain
2478:
2393:
3274:
2375:
2370:
3406:
3366:
3168:
3160:
2904:
2787:
2616:
Henri Milne - Edwards's (1844) diagram of relationships of vertebrate animals, illustrated as a series of nested sets
354:
74:
3244:
2421:
2386:
2368:
863:". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems":
2382:
2166:
is a tautology, "truth" is always the case no matter how x, y and z are valued, but "truth" is only the case for
2062:(i.e. ~(y & z) & (x → y) ) → ~(x & z) ) is still a formula, and the deduction – the "detachment" of
144:, that have common elements; the zone inside both curves represents the set of elements common to both sets (the
2871:(NB. Has a detailed history of the evolution of logic diagrams including but not limited to the Euler diagram.)
2403:
357:
erroneously asserted that the original use of the circles to "sensualize... the abstractions of logic" was not
2420:
2428:
2404:
2385:
2219:
i.e.: ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) , ( ~(y & z) & (x → y) ) ⊢ ( ~ (x & z) )
533:“It fits in, but badly, even with the four propositions of the common logic to which it is normally applied.”
2424:
108:(1707–1783). In the United States, both Venn and Euler diagrams were incorporated as part of instruction in
2956:
367:
598:
from the “... old-fashioned Euler diagrams.” In particular Venn gives an example, shown at the left.
3529:
3514:
2401:
1202:
This example shows the Euler and Venn diagrams and
Karnaugh map deriving and verifying the deduction "No
2411:
2410:
2299:
s"; the criterion for a successful deduction is that the 1s under the sub-major connective on the right
3452:
3340:
3184:
2396:
2395:
2372:
3439:
3220:
Transactions of the
American Institute of Electrical Engineers, Part I: Communication and Electronics
2856:
2419:
2413:
2409:
2408:
995:
605:(1868–1914) had labeled the terms as shown on the drawing at the right. Moreover, he had labeled the
145:
141:
2434:
2312:" has all the same 1s that appear in the bold-faced column under the left-side sub-major connective
1358:. If the evaluation of the truth table produces all 1s under the implication-sign (→, the so-called
19:
This article is about Eulerian circles of set theory and logic. For the geometric Euler circle, see
3398:
3358:
3172:
2952:
2686:
2430:
2427:
2425:
2423:
2402:
2397:
2394:
2378:
2432:
2418:
2414:
2406:
2399:
1150:
In a logical setting, one can use model-theoretic semantics to interpret Euler diagrams, within a
3504:
3200:
2679:
By the time these lectures of Hamilton were published, Hamilton had died. His editors (marked by
2539:, using the definition that isosceles triangles have at least (rather than exactly) 2 equal sides
2391:
2390:
2387:
2100:(or "the fundamental rule of inference") is often written as follows: The two terms on the left,
1225:
1082:
1042:
375:
175:
3349:
1921 "Introduction to a general theory of elementary propositions" reprinted with commentary by
2433:
2431:
2417:
2398:
1265:
Before it can be presented in a Venn diagram or Karnaugh Map, the Euler diagram's syllogism "No
3328:
2412:
2407:
2405:
2400:
2392:
1282:
727:
In Venn the background surrounding the circles, does not appear: That is, the term marked "0",
2445:
showing the relationships between various multinational European organisations and agreements
2389:
2388:
911:
In Chapter 6, section 6.4 "Karnaugh map representation of Boolean functions" they begin with:
2739:
1151:
132:
of the set, and the exterior, which represents all elements that are not members of the set.
129:
124:
1127:
3489:
3417:
2291:
The use of tautological implication means that other possible deductions exist besides "No
1139:
943:. In Veitch's method the variables are arranged in a rectangle or square; as described in
8:
3350:
2948:
2852:
2706:
2655:
1339:
1323:
1186:
of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent
840:
510:
358:
3307:
3236:
3176:
2690:
34:
2819:
2581:
Euler Diagram displaying the relationship between Homographs, homophones, and synonyms
227:
3564:
3421:
3402:
3372:
3362:
3192:
3115:
2825:
2639:
1371:
87:
20:
3311:
3240:
1261:
94:
and overlapping definitions. They are similar to another set diagramming technique,
3524:
3390:
3332:
3299:
3287:
3228:
3207:
2737:
This is a sophisticated concept. Russell and Whitehead (2nd edition 1927) in their
1306:
876:
872:
53:
3494:
2922:
2288:
s", perhaps to use it in a subsequent deduction (or as a topic of conversation).
362:
1162:
are disjoint since the corresponding curves are disjoint, and also that the set
887:
and the application of propositional logic to switching logic by (among others)
140:, which have no elements in common. Two curves that overlap represent sets that
3152:
2719:
2649:
1330:. The easiest method is put the starting formula on the left (abbreviate it as
892:
888:
602:
120:
105:
2791:
2500:
Euler diagram visualizing a real situation, the relationships between various
641:
the circles can be summed to yield the following equation for Venn's example:
517:
3553:
3232:
2567:
2552:
1374:. Given this fact, one can "detach" the formula on the right (abbreviated as
595:
577:“We now come to Euler's well-known circles which were first described in his
137:
102:
3096:
Hill & Peterson (1968) . "Set theory as an example of Boolean algebra".
974:, showing how they can be easily transformed into equivalent Euler diagrams
3499:
3484:
2521:
2485:
2083:
981:
964:
944:
155:
95:
39:
3303:
2652:– an extension of Euler diagrams adding existence to contour intersections
1210:
s". In the illustration and table the following logical symbols are used:
3519:
2644:
1381:
Given the example above, the formula for the Euler and Venn diagrams is:
1327:
1302:
959:
896:
880:
163:
3539:
505:(1834–1923) comments on the remarkable prevalence of the Euler diagram:
329:
3384:
2660:
1217:~ for NOT and abbreviated to ' when illustrating the minterms e.g. x' =
723:
the circles (but this is not entirely correct; see the next paragraph).
109:
3292:
Proceedings of the 1952 ACM national meeting (Pittsburgh) on - ACM '52
3346:
3073:
3054:
3012:
2815:
971:
948:
884:
847:
621:') as well. He succinctly explains how to use the diagram – one must
538:
502:
379:
216:
148:
of the sets). A curve completely within the interior of another is a
91:
90:
and their relationships. They are particularly useful for explaining
3444:
101:
The first use of "Eulerian circles" is commonly attributed to Swiss
3355:
From Frege to Gödel: A Source Book of Mathematical Logic, 1879–1931
2593:
The 22 (of 256) essentially different Venn diagrams with 3 circles
2536:
2082:
is tautology, the stage is now set for the use of the procedure of
1197:
334:
113:
2685:.), responsible for most of the footnote text, were the logicians
1334:) and put the (possible) deduction on the right (abbreviate it as
1238:→ (logical IMPLICATION): read as IF ... THEN ..., or " IMPLIES ",
215:
refer to four types of categorical statement which can occur in a
3290:(1952-05-03) . "A chart method for simplifying truth functions".
1154:. In the examples below, the Euler diagram depicts that the sets
233:
83:
3123:
149:
26:
3212:"The Map Method for Synthesis of Combinational Logic Circuits"
1281:" must first be reworded into the more formal language of the
1182:, does not encapsulate these relationships. Traditionally the
65:
59:
2673:
257:
is represented by arithmetic multiplication, and the logical
133:
3375:
1938 "A Symbolic Analysis of Relay and Switching Circuits",
265:
is represented by " ' " after the variable, e.g. the region
3078:
Introduction to a general theory of elementary propositions
2974:
2972:
2303:
all the 1s under the sub-major connective on the left (the
954:
633:
those which are made to vanish by the data of the problem."
3091:
3089:
3087:
1433:
So now the formula to be evaluated can be abbreviated to:
939:, in turn referenced (among other authors of logic texts)
38:
Euler diagram showing the relationships between different
16:
Graphical set representation involving overlapping circles
3015:(1881a). "Chapter V – Diagrammatic representation".
249:
per Venn's drawing. The symbolism is as follows: logical
2969:
875:(1924–2022) would adapt and expand a method proposed by
3377:
Transactions American Institute of Electrical Engineers
3084:
1170:
s. The Venn diagram, which uses the same categories of
637:
Given the Venn's assignments, then, the unshaded areas
333:
Both the Veitch diagram and Karnaugh map show all the
2074:– has not occurred. But given the demonstration that
1085:
1045:
998:
75:
3412:
Frederich J. Hill and Gerald R. Peterson 1968, 1974
3040:, p. 100 of "old-fashioned Eulerian diagrams"
2788:"Strategies for Reading Comprehension Venn Diagrams"
2757:
Reichenbach discusses the fact that the implication
1437:( ~(y & z) & (x → y) ) → ( ~ (x & z) ):
62:
3414:
Introduction to Switching Theory and Logical Design
1121:The Euler and the Venn diagrams of those sets are:
237:. They are shown in the diagram with the variables
56:
1110:
1070:
1030:
2943:
2941:
2897:
1378:) in the manner described below the truth table.
1190:either by shading or by the absence of a region.
3551:
3007:
3005:
3003:
3001:
2999:
2997:
2995:
2993:
2991:
2951:(1842) . "Partie II, Lettre XXXV". In
2198:: x'y'z' + x'y'z + x'yz' + xyz' = x'y' + yz').
1198:Example: Euler- to Venn-diagram and Karnaugh map
2366:
2280:One is now free to "detach" the conclusion "No
2142:For the modus ponens to succeed, both premises
1322:", one can test whether or not it is a correct
3047:
2938:
2887:
2885:
2883:
2881:
2879:
2877:
2837:
2835:
2566:Euler diagram categorizing different types of
3460:
3057:(1881b). "Chapter XX – Historic notes".
3036:, pp. 114 ff; in the "Eulerian scheme"
2988:
2935:— Published 4 years after Weise's death.
2903:
2457:
2158:. Because, as demonstrated above the premise
2094:s" and dispense with the terms on the left.
3383:edited by N.J.A. Solane and Aaron D. Wyner,
3095:
1105:
1092:
1065:
1052:
1025:
1005:
220:
2874:
2832:
2810:
2808:
871:Thus the matter would rest until 1952 when
353:As shown in the illustration to the right,
3467:
3453:
2464:
2450:
2841:
1101:
1061:
1021:
1014:
3261:
3206:
3167:
3151:
3030:
2978:
2891:
2805:
2702:
1314:Given a proposed conclusion such as "No
1260:
958:
955:Relation between Euler and Venn diagrams
940:
924:
516:
328:
226:
174:
33:
25:
3381:Claude Elwood Shannon: Collected Papers
3379:vol 57, pp. 471–495. Derived from
3114:
2981:, p. 179; these examples are from
2597:and their corresponding Euler diagrams.
936:
932:
374:In Hamilton's illustration of the four
3552:
3286:
3191:
3120:: In effect, Shannon's master's thesis
2982:
928:
3474:
3448:
3053:
3041:
3037:
3033:
3011:
2947:
2921:
2520:Humorous diagram comparing Euler and
1214:1 can be read as "true", 0 as "false"
522:
3510:Propositional directed acyclic graph
3438:Euler Diagrams. Brighton, UK (2004).
3353:in Jean van Heijenoort, editor 1967
3072:
2965:] (in French). pp. 412–417.
2814:
2551:Euler diagram of terminology of the
2502:supranational European organizations
2054:At this point the above implication
1401:s": ( ~(y & z) & (x → y) ) =
846:Couturat observed that, in a direct
454:"No metals are compound substances."
3335:1913 1st edition, 1927 2nd edition
3296:Association for Computing Machinery
2958:Lettres a une Princesse d'Allemagne
2842:Mac Queen, Gailand (October 1967).
579:Lettres a une Princesse d'Allemagne
473:Example: "Some metals are brittle."
13:
3319:
2628:Euler diagram of numbers under 100
2488:showing all possible intersections
2362:
14:
3576:
3432:
3181:Lectures on Metaphysics and Logic
3161:The Open Court Publishing Company
2909:Lectures on Metaphysics and Logic
970:with shaded regions representing
3560:Graphical concepts in set theory
3538:
2621:
2609:
2586:
2574:
2559:
2544:
2528:
2513:
2493:
2477:
1138:
1126:
625:the regions that are to vanish:
52:
3129:
3108:
3066:
3023:
2751:
2731:
2712:
2696:
2174:evaluates as "true" (e.g. rows
1411:And the proposed deduction is:
1031:{\displaystyle A=\{1,\,2,\,5\}}
136:which do not overlap represent
3243:. Paper 53-217. Archived from
2915:
2780:
1285:: " 'It is not the case that:
879:; this work would rely on the
495:"Some metals are not brittle."
382:as symbolized by the drawings
1:
3262:Sandifer, Ed (January 2004).
2773:
3343:(1962 edition), UK, no ISBN.
3337:Principia Mathematica to *56
2963:Letters to a German Princess
2170:in those circumstances when
883:method precisely defined by
521:Composite of two pages from
368:Letters to a German Princess
7:
3530:Method of analytic tableaux
3515:Sentential decision diagram
2855:. p. 5. Archived from
2633:
2348:
2342:
2336:
2330:
2324:
2318:
2194:
2188:
2182:
2176:
1989:
1925:
1861:
1797:
1733:
1669:
1605:
1541:
1338:) and connect the two with
1111:{\displaystyle C=\{4,\,7\}}
1071:{\displaystyle B=\{1,\,6\}}
10:
3581:
3395:Elements of Symbolic Logic
3185:William Blackwood and Sons
3145:
3019:. p. 100, Footnote 1.
2535:Euler diagram of types of
2355:
1166:is a subset of the set of
433:"All metals are elements."
170:
119:Euler diagrams consist of
18:
3536:
3480:
3173:Mansel, Henry Longueville
2927:Nucleus Logicae Weisianae
2746:Principles of Mathematics
162:curves, representing all
3440:What are Euler Diagrams?
3399:Dover Publications, Inc.
3359:Harvard University Press
3288:Veitch, Edward Westbrook
3233:10.1109/TCE.1953.6371932
3183:. Edinburgh and London:
2687:Henry Longueville Mansel
2666:
376:categorical propositions
3505:Binary decision diagram
3324:By date of publishing:
3201:M. A. MacMillan and Co.
3199:. London and New York:
2340:), plus two more (rows
1423:s": ( ~ (x & z) ) =
1224:+ for Boolean OR (from
631:strike out (by shading)
361:(1707–1783) but rather
349:are per Venn's example.
179:A page from Hamilton's
3329:Alfred North Whitehead
3159:. Chicago and London:
3135:cf Reichenbach 1947:64
2931:Weissian core of logic
2437:
1484:Venn, Karnaugh region
1311:
1283:propositional calculus
1112:
1072:
1032:
978:
719:for the unshaded area
573:) are simply rotated:
571:Particular Affirmative
526:
469:Particular Affirmative
350:
326:
224:
86:means of representing
43:
31:
3418:John Wiley & Sons
3304:10.1145/609784.609801
2740:Principia Mathematica
2436:
1264:
1152:universe of discourse
1113:
1073:
1033:
962:
931:, Veitch, referenced
520:
427:Universal Affirmative
378:which can occur in a
332:
230:
178:
125:two-dimensional plane
37:
29:
3490:Square of opposition
3397:republished 1980 by
3298:. pp. 127–133.
3169:Sir William Hamilton
1083:
1043:
996:
355:Sir William Hamilton
121:simple closed shapes
3351:Jean van Heijenoort
2853:McMaster University
2656:Three circles model
2086:to "detach" Q: "No
1477:
1340:logical implication
841:domain of discourse
785:" has the equation
684:" has the equation
559:Particular Negative
489:Particular Negative
92:complex hierarchies
3264:"How Euler Did It"
3210:(November 1953) .
3193:Jevons, W. Stanley
2985:, pp. 71 ff.
2894:, pp. 73, 75
2438:
1475:
1312:
1108:
1068:
1028:
979:
963:Examples of small
527:
448:Universal Negative
351:
327:
225:
221:descriptions, left
181:Lectures on Logic;
44:
32:
3547:
3546:
3475:Diagrams in logic
3426:978-0-471-39882-0
3373:Claude E. Shannon
3361:, Cambridge, MA,
3339:Cambridge At The
3294:. New York, USA:
3208:Karnaugh, Maurice
2933:] (in Latin).
2845:The Logic Diagram
2826:MacMillan and Co.
2640:Intersectionality
2604:
2506:clickable version
2052:
2051:
21:Nine-point circle
3572:
3542:
3525:Sequent calculus
3469:
3462:
3455:
3446:
3445:
3391:Hans Reichenbach
3341:University Press
3333:Bertrand Russell
3315:
3281:
3279:
3273:. Archived from
3268:
3258:
3256:
3255:
3249:
3216:
3203:
3188:
3164:
3136:
3133:
3127:
3126:
3121:
3112:
3106:
3105:
3103:
3093:
3082:
3081:
3070:
3064:
3062:
3051:
3045:
3027:
3021:
3020:
3009:
2986:
2976:
2967:
2966:
2945:
2936:
2934:
2919:
2913:
2912:
2901:
2895:
2889:
2872:
2870:
2868:
2867:
2861:
2850:
2839:
2830:
2829:
2812:
2803:
2802:
2800:
2799:
2790:. Archived from
2784:
2767:
2755:
2749:
2735:
2723:
2718:See footnote in
2716:
2710:
2705:points out that
2700:
2694:
2684:
2683:
2677:
2625:
2613:
2601:
2590:
2578:
2563:
2548:
2532:
2517:
2497:
2481:
2466:
2459:
2452:
2365:
1991:
1927:
1863:
1799:
1735:
1671:
1607:
1543:
1478:
1474:
1463:s" ) THEN ( "No
1360:major connective
1307:Boolean equation
1142:
1130:
1117:
1115:
1114:
1109:
1077:
1075:
1074:
1069:
1037:
1035:
1034:
1029:
877:Edward W. Veitch
873:Maurice Karnaugh
862:
858:
855:
854:
835:
831:
827:
823:
818:
814:
810:
806:
803:
800:
796:
793:
789:
784:
780:
777:
776:
771:
767:
764:
763:
758:
754:
751:
750:
741:
739:
735:
731:
718:
717:
713:
709:
705:
702:
699:
695:
692:
688:
683:
679:
676:
675:
670:
666:
663:
662:
657:
653:
650:
649:
620:
616:
612:
567:
566:
555:
554:
485:
484:
465:
464:
444:
443:
423:
422:
412:
411:
404:
403:
396:
395:
388:
387:
348:
344:
340:
325:
323:
319:
315:
309:
306:
305:
300:
297:
296:
291:
290:
285:
282:
281:
276:
272:
268:
264:
263:
262:
256:
255:
254:
248:
244:
240:
213:
212:
205:
204:
197:
196:
189:
188:
78:
72:
71:
68:
67:
64:
61:
58:
3580:
3579:
3575:
3574:
3573:
3571:
3570:
3569:
3550:
3549:
3548:
3543:
3534:
3495:Porphyrian tree
3476:
3473:
3435:
3322:
3320:Further reading
3277:
3266:
3253:
3251:
3247:
3214:
3153:Couturat, Louis
3148:
3142:
3140:
3139:
3134:
3130:
3119:
3113:
3109:
3102:sections 4.5 ff
3101:
3098:Boolean Algebra
3094:
3085:
3080:(Ph.D. thesis).
3071:
3067:
3052:
3048:
3031:Sandifer (2004)
3028:
3024:
3010:
2989:
2979:Hamilton (1860)
2977:
2970:
2946:
2939:
2920:
2916:
2902:
2898:
2892:Couturat (1914)
2890:
2875:
2865:
2863:
2859:
2848:
2840:
2833:
2813:
2806:
2797:
2795:
2786:
2785:
2781:
2776:
2771:
2770:
2756:
2752:
2736:
2732:
2727:
2726:
2717:
2713:
2703:Sandifer (2004)
2701:
2697:
2681:
2680:
2678:
2674:
2669:
2636:
2629:
2626:
2617:
2614:
2605:
2602:
2591:
2582:
2579:
2570:
2564:
2555:
2549:
2540:
2533:
2524:
2518:
2509:
2498:
2489:
2482:
2473:
2472:
2471:
2470:
2439:
2435:
2363:
2358:
1426:
1404:
1300:
1296:
1292:
1288:
1249:
1234:
1226:Boolean algebra
1220:
1200:
1146:
1143:
1134:
1131:
1084:
1081:
1080:
1044:
1041:
1040:
997:
994:
993:
957:
941:Couturat (1914)
925:Karnaugh (1953)
860:
856:
852:
851:
829:
825:
821:
816:
812:
808:
804:
801:
798:
794:
791:
787:
786:
782:
778:
774:
773:
769:
765:
761:
760:
756:
752:
748:
747:
737:
733:
729:
728:
715:
711:
707:
703:
700:
697:
693:
690:
686:
685:
681:
677:
673:
672:
668:
664:
660:
659:
655:
651:
647:
646:
618:
614:
610:
607:exterior region
564:
563:
552:
551:
482:
481:
462:
461:
441:
440:
420:
419:
409:
408:
401:
400:
393:
392:
385:
384:
346:
342:
338:
321:
317:
313:
311:
307:
303:
302:
298:
294:
293:
288:
287:
283:
279:
278:
274:
270:
266:
260:
259:
258:
252:
251:
250:
246:
242:
238:
210:
209:
202:
201:
194:
193:
186:
185:
173:
112:as part of the
76:
55:
51:
24:
17:
12:
11:
5:
3578:
3568:
3567:
3562:
3545:
3544:
3537:
3535:
3533:
3532:
3527:
3522:
3517:
3512:
3507:
3502:
3497:
3492:
3487:
3481:
3478:
3477:
3472:
3471:
3464:
3457:
3449:
3443:
3442:
3434:
3433:External links
3431:
3430:
3429:
3410:
3388:
3370:
3344:
3321:
3318:
3317:
3316:
3283:
3282:
3280:on 2013-01-26.
3259:
3227:(5): 593–599.
3204:
3189:
3165:
3147:
3144:
3138:
3137:
3128:
3107:
3083:
3065:
3061:. p. 424.
3059:Symbolic Logic
3046:
3044:, p. 113
3022:
3017:Symbolic Logic
2987:
2968:
2937:
2914:
2911:. p. 180.
2905:Hamilton, W.R.
2896:
2873:
2831:
2821:Symbolic Logic
2804:
2778:
2777:
2775:
2772:
2769:
2768:
2750:
2729:
2728:
2725:
2724:
2720:George Stibitz
2711:
2695:
2671:
2670:
2668:
2665:
2664:
2663:
2658:
2653:
2650:Spider diagram
2647:
2642:
2635:
2632:
2631:
2630:
2627:
2620:
2618:
2615:
2608:
2606:
2592:
2585:
2583:
2580:
2573:
2571:
2568:metaheuristics
2565:
2558:
2556:
2550:
2543:
2541:
2534:
2527:
2525:
2519:
2512:
2510:
2499:
2492:
2490:
2483:
2476:
2469:
2468:
2461:
2454:
2446:
2440:
2361:
2360:
2359:
2357:
2354:
2278:
2277:
2276:
2275:
2220:
2140:
2139:
2050:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1986:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1922:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1858:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1794:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1730:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1666:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1602:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1538:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1496:
1493:
1490:
1487:
1485:
1482:
1473:
1472:
1445:
1431:
1430:
1424:
1409:
1408:
1402:
1298:
1294:
1290:
1286:
1259:
1258:
1247:
1236:
1232:
1229:
1222:
1218:
1215:
1199:
1196:
1148:
1147:
1144:
1137:
1135:
1132:
1125:
1119:
1118:
1107:
1104:
1100:
1097:
1094:
1091:
1088:
1078:
1067:
1064:
1060:
1057:
1054:
1051:
1048:
1038:
1027:
1024:
1020:
1017:
1013:
1010:
1007:
1004:
1001:
956:
953:
937:Shannon (1938)
933:Shannon (1938)
921:
920:
919:
918:
909:
908:
907:
906:
869:
868:
837:
836:
725:
724:
635:
634:
592:
591:
547:
546:
535:
534:
515:
514:
500:
499:
498:
497:
478:
477:
476:
458:
457:
456:
437:
436:
435:
172:
169:
106:Leonhard Euler
15:
9:
6:
4:
3:
2:
3577:
3566:
3563:
3561:
3558:
3557:
3555:
3541:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3482:
3479:
3470:
3465:
3463:
3458:
3456:
3451:
3450:
3447:
3441:
3437:
3436:
3427:
3423:
3419:
3415:
3411:
3408:
3407:0-486-24004-5
3404:
3400:
3396:
3392:
3389:
3386:
3382:
3378:
3374:
3371:
3368:
3367:0-674-32449-8
3364:
3360:
3356:
3352:
3348:
3345:
3342:
3338:
3334:
3330:
3327:
3326:
3325:
3313:
3309:
3305:
3301:
3297:
3293:
3289:
3285:
3284:
3276:
3272:
3265:
3260:
3250:on 2017-04-16
3246:
3242:
3238:
3234:
3230:
3226:
3222:
3221:
3213:
3209:
3205:
3202:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3154:
3150:
3149:
3143:
3132:
3125:
3117:
3116:Shannon, C.E.
3111:
3099:
3092:
3090:
3088:
3079:
3075:
3069:
3060:
3056:
3050:
3043:
3039:
3035:
3032:
3026:
3018:
3014:
3008:
3006:
3004:
3002:
3000:
2998:
2996:
2994:
2992:
2984:
2983:Jevons (1880)
2980:
2975:
2973:
2964:
2960:
2959:
2954:
2950:
2944:
2942:
2932:
2928:
2924:
2918:
2910:
2907:(1858–1860).
2906:
2900:
2893:
2888:
2886:
2884:
2882:
2880:
2878:
2862:on 2017-04-14
2858:
2854:
2847:
2846:
2838:
2836:
2827:
2823:
2822:
2817:
2811:
2809:
2794:on 2009-04-29
2793:
2789:
2783:
2779:
2764:
2760:
2754:
2747:
2742:
2741:
2734:
2730:
2721:
2715:
2708:
2704:
2699:
2692:
2688:
2676:
2672:
2662:
2659:
2657:
2654:
2651:
2648:
2646:
2643:
2641:
2638:
2637:
2624:
2619:
2612:
2607:
2600:
2596:
2589:
2584:
2577:
2572:
2569:
2562:
2557:
2554:
2553:British Isles
2547:
2542:
2538:
2531:
2526:
2523:
2522:Venn diagrams
2516:
2511:
2507:
2503:
2496:
2491:
2487:
2480:
2475:
2474:
2467:
2462:
2460:
2455:
2453:
2448:
2447:
2444:
2443:Euler diagram
2353:
2351:
2350:
2345:
2344:
2339:
2338:
2333:
2332:
2327:
2326:
2321:
2320:
2315:
2311:
2306:
2302:
2298:
2294:
2289:
2287:
2283:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2233:
2229:
2225:
2222:i.e.: IF "No
2221:
2218:
2217:
2216:
2212:
2208:
2204:
2201:
2200:
2199:
2197:
2196:
2191:
2190:
2185:
2184:
2179:
2178:
2173:
2169:
2165:
2161:
2157:
2153:
2149:
2145:
2138:
2134:
2130:
2126:
2123:
2122:
2121:
2119:
2115:
2112:, are called
2111:
2107:
2103:
2099:
2095:
2093:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2057:
2047:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1988:
1987:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1924:
1923:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1860:
1859:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1796:
1795:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1732:
1731:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1668:
1667:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1604:
1603:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1540:
1539:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1497:
1494:
1491:
1488:
1486:
1483:
1480:
1479:
1470:
1466:
1462:
1458:
1454:
1450:
1446:
1444:
1440:
1436:
1435:
1434:
1429:
1422:
1418:
1414:
1413:
1412:
1407:
1400:
1396:
1392:
1388:
1384:
1383:
1382:
1379:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1350:, read as IF
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1308:
1304:
1284:
1280:
1276:
1272:
1268:
1263:
1257:
1253:
1245:
1241:
1237:
1230:
1227:
1223:
1216:
1213:
1212:
1211:
1209:
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1191:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1141:
1136:
1133:Euler diagram
1129:
1124:
1123:
1122:
1102:
1098:
1095:
1089:
1086:
1079:
1062:
1058:
1055:
1049:
1046:
1039:
1022:
1018:
1015:
1011:
1008:
1002:
999:
992:
991:
990:
987:
983:
982:Venn diagrams
977:
973:
969:
966:
965:Venn diagrams
961:
952:
950:
946:
942:
938:
934:
930:
929:Veitch (1952)
926:
916:
915:
914:
913:
912:
904:
903:
902:
901:
900:
898:
894:
890:
886:
882:
878:
874:
866:
865:
864:
849:
844:
842:
833:
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744:
743:
722:
644:
643:
642:
640:
632:
628:
627:
626:
624:
608:
604:
599:
597:
596:Venn diagrams
589:
584:
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576:
575:
574:
572:
568:
560:
556:
544:
543:
542:
540:
532:
531:
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524:
519:
512:
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506:
504:
496:
492:
491:
490:
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479:
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471:
470:
466:
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451:
450:
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445:
438:
434:
430:
429:
428:
424:
417:
416:
415:
413:
405:
397:
389:
381:
377:
372:
370:
369:
364:
360:
356:
336:
331:
277:is read as "(
236:
235:
229:
222:
218:
214:
206:
198:
190:
182:
177:
168:
165:
161:
157:
156:Venn diagrams
153:
151:
147:
143:
139:
138:disjoint sets
135:
131:
126:
122:
117:
115:
111:
107:
104:
103:mathematician
99:
97:
96:Venn diagrams
93:
89:
85:
81:
80:
70:
49:
48:Euler diagram
41:
36:
28:
22:
3500:Karnaugh map
3485:Venn diagram
3413:
3394:
3380:
3376:
3354:
3336:
3323:
3291:
3275:the original
3270:
3252:. Retrieved
3245:the original
3224:
3218:
3196:
3180:
3177:Veitch, John
3156:
3141:
3131:
3110:
3097:
3077:
3068:
3058:
3049:
3042:Venn (1881a)
3038:Venn (1881a)
3034:Venn (1881a)
3025:
3016:
2962:
2957:
2930:
2926:
2917:
2908:
2899:
2864:. Retrieved
2857:the original
2844:
2828:p. 509.
2820:
2796:. Retrieved
2792:the original
2782:
2762:
2758:
2753:
2745:
2738:
2733:
2714:
2698:
2675:
2598:
2594:
2486:Venn diagram
2442:
2347:
2341:
2335:
2329:
2323:
2317:
2313:
2309:
2304:
2300:
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2292:
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2279:
2271:
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2259:
2258:s" and "All
2255:
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2247:
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2227:
2223:
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2210:
2206:
2202:
2193:
2187:
2181:
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2167:
2163:
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2151:
2147:
2143:
2141:
2136:
2132:
2128:
2124:
2117:
2113:
2109:
2105:
2101:
2098:Modus ponens
2097:
2096:
2091:
2087:
2084:modus ponens
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2053:
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1390:
1386:
1380:
1375:
1367:
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1359:
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1351:
1347:
1343:
1335:
1331:
1326:by use of a
1319:
1315:
1313:
1278:
1274:
1270:
1266:
1255:
1251:
1243:
1239:
1207:
1203:
1201:
1192:
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1159:
1155:
1149:
1145:Venn diagram
1120:
985:
980:
975:
967:
945:Karnaugh map
922:
910:
870:
845:
838:
820:
772:: therefore
726:
720:
671:: therefore
638:
636:
630:
622:
606:
600:
593:
587:
582:
578:
570:
562:
558:
550:
548:
536:
528:
523:Venn (1881a)
501:
494:
488:
480:
474:
468:
460:
453:
447:
439:
432:
426:
418:
407:
399:
391:
383:
373:
366:
352:
232:
208:
200:
192:
184:
183:the symbols
180:
164:combinations
159:
154:
146:intersection
118:
100:
84:diagrammatic
47:
45:
40:Solar System
3520:Truth table
3387:, New York.
2949:Euler, L.P.
2766:1947:64–66.
2691:John Veitch
2645:Rainbow box
1481:Square no.
1328:truth table
1303:truth table
1293:AND 'If an
927:referenced
881:truth table
848:algorithmic
539:algorithmic
316:) & (¬
3554:Categories
3385:IEEE Press
3254:2017-04-16
3122:(Report).
2866:2017-04-14
2851:(Thesis).
2824:. London:
2816:Venn, John
2798:2009-06-20
2774:References
2661:UpSet plot
2118:conclusion
972:empty sets
623:strike out
609:(shown as
541:practice:
110:set theory
3347:Emil Post
2923:Weise, C.
2537:triangles
2266:s" ⊢ "No
1447:IF ( "No
1372:tautology
1324:deduction
1188:emptiness
1184:emptiness
1180:Four Legs
1164:Four Legs
968:(on left)
949:hypercube
885:Emil Post
601:By 1914,
493:Example:
452:Example:
431:Example:
380:syllogism
217:syllogism
142:intersect
3565:Diagrams
3312:17284651
3241:51636736
3195:(1880).
3179:(eds.).
3171:(1860).
3155:(1914).
3118:(1938).
3076:(1921).
3074:Post, E.
3055:Venn, J.
3013:Venn, J.
2925:(1712).
2818:(1881).
2722:article.
2634:See also
2599:(bottom)
2250:s", "No
2154:must be
2114:premises
1310:example.
603:Couturat
513:scheme."
511:Eulerian
335:minterms
320:) &
234:minterms
130:elements
114:new math
3271:maa.org
3146:Sources
2955:(ed.).
2953:Cournot
2356:Gallery
2301:include
2066:out of
2009:
1997:
1945:
1933:
1881:
1869:
1817:
1805:
1753:
1741:
1689:
1677:
1625:
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1561:
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1546:x'y'z'
1425:defined
1403:defined
1362:) then
1297:then a
1248:defined
1246:=
1233:defined
1219:defined
1176:Mineral
1160:Mineral
976:(right)
893:Stibitz
889:Shannon
583:Letters
310:" i.e.
171:History
152:of it.
82:) is a
42:objects
3424:
3420:, NY,
3405:
3401:, NY,
3369:(pbk.)
3365:
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3239:
3124:M.I.T.
2316:(rows
2295:s are
2284:s are
2270:s are
2262:s are
2254:s are
2246:s are
2234:s are
2226:s are
2090:s are
1802:xy'z'
1674:x'yz'
1610:x'y'z
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1467:s are
1459:s are
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1419:s are
1397:s are
1389:s are
1273:, All
1221:NOT x,
1206:s are
1178:, and
1172:Animal
1168:Animal
1156:Animal
935:, and
897:Turing
895:, and
721:inside
639:inside
561:) and
487:: The
467:: The
446:: The
425:: The
406:, and
345:, and
245:, and
207:, and
150:subset
134:Curves
3393:1947
3308:S2CID
3278:(PDF)
3267:(PDF)
3248:(PDF)
3237:S2CID
3215:(PDF)
2961:[
2929:[
2860:(PDF)
2849:(PDF)
2707:Euler
2667:Notes
2595:(top)
2314:&
2305:major
1930:xyz'
1866:xy'z
1738:x'yz
1370:is a
1354:THEN
1342:i.e.
1318:is a
414:are:
363:Weise
359:Euler
219:(see
123:in a
3422:ISBN
3403:ISBN
3363:ISBN
3331:and
2748:§38.
2689:and
2346:and
2334:and
2242:"No
2240:THEN
2156:true
2150:and
2108:and
1994:xyz
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1521:y))
1471:s" )
1415:"No
1385:"No
1289:AND
1250:NOT
1158:and
807:' +
797:' +
759:and
706:' +
696:' +
658:and
503:Venn
88:sets
79:-lər
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2192:OR
2186:OR
2180:OR
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1527:(~
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762:ALL
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304:AND
295:NOT
289:AND
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3357:,
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3269:.
3235:.
3225:72
3223:.
3217:.
3175:;
3100:.
3086:^
2990:^
2971:^
2940:^
2876:^
2834:^
2807:^
2761:→
2682:ED
2484:A
2328:,
2322:,
2274:s"
2213:⊢
2209:,
2205:→
2162:→
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2135:⊢
2131:,
2127:→
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2078:→
2070:→
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2045:1
2042:1
2039:0
2036:1
2033:1
2030:1
2027:1
2024:0
2021:1
2018:1
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2006:1
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2000:1
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1981:0
1978:1
1975:1
1972:1
1969:1
1966:1
1963:1
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951:.
891:,
853:NO
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674:NO
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