Knowledge

Projection (set theory)

Source 📝

651: 397:
from a set to another set. The function from elements to equivalence classes is a surjection, and every surjection corresponds to an equivalence relation under which two elements are equivalent when they have the same image. The result of the mapping is written as
267: 193: 339: 95: 475: 391: 495: 442: 361: 54: 422: 201: 100: 692: 711: 272: 567: 629: 602: 575: 530: 685: 59: 524: 519: 716: 513: 678: 666: 536: 25: 619: 561: 592: 368: 447: 8: 373: 480: 427: 346: 39: 401: 625: 598: 571: 507: 364: 196: 662: 705: 533: – Operation that restricts a relation to a specified set of attributes 262:{\displaystyle (X_{1}\times \cdots \times X_{j}\times \cdots \times X_{k})} 557: 539: – Relationship between two sets, defined by a set of ordered pairs 658: 394: 33: 17: 527: – Idempotent linear transformation from a vector space to itself 597:, Graduate Texts in Mathematics, vol. 154, Springer, p. 8, 188:{\displaystyle {\vec {x}}=(x_{1},\ \dots ,\ x_{j},\ \dots ,\ x_{k})} 650: 516: – Mapping equal to its square under mapping composition 624:, Springer Monographs in Mathematics, Springer, p. 34, 334:{\displaystyle \mathrm {proj} _{j}({\vec {x}})=x_{j}.} 483: 450: 430: 404: 376: 349: 275: 204: 103: 62: 42: 510: – Mathematical set formed from two given sets 489: 469: 436: 416: 385: 355: 333: 261: 187: 89: 48: 703: 686: 590: 693: 679: 621:Set Theory: The Third Millennium Edition 24:is one of two closely related types of 704: 591:Brown, Arlen; Pearcy, Carl M. (1995), 556: 645: 617: 90:{\displaystyle \mathrm {proj} _{j},} 13: 568:Undergraduate Texts in Mathematics 287: 284: 281: 278: 74: 71: 68: 65: 14: 728: 343:A function that sends an element 649: 531:Projection (relational algebra) 611: 584: 550: 458: 451: 411: 405: 312: 306: 297: 256: 205: 182: 119: 110: 1: 543: 477:when it is necessary to make 444:is understood, or written as 712:Basic concepts in set theory 665:. You can help Knowledge by 7: 594:An Introduction to Analysis 525:Projection (linear algebra) 520:Projection (measure theory) 501: 10: 733: 644: 36:operation typified by the 570:, Springer, p. 32, 514:Projection (mathematics) 56:projection map, written 28:or operations, namely: 661:-related article is a 537:Relation (mathematics) 491: 471: 438: 418: 387: 357: 335: 263: 189: 97:that takes an element 91: 50: 618:Jech, Thomas (2003), 492: 472: 439: 419: 388: 358: 336: 264: 190: 92: 51: 481: 470:{\displaystyle _{E}} 448: 428: 402: 393:or, equivalently, a 374: 369:equivalence relation 347: 273: 202: 101: 60: 40: 487: 467: 434: 414: 386:{\displaystyle E,} 383: 367:under a specified 353: 331: 259: 185: 87: 46: 674: 673: 508:Cartesian product 490:{\displaystyle E} 437:{\displaystyle E} 365:equivalence class 356:{\displaystyle x} 309: 197:Cartesian product 171: 162: 146: 137: 113: 49:{\displaystyle j} 724: 717:Set theory stubs 695: 688: 681: 653: 646: 636: 634: 615: 609: 607: 588: 582: 580: 563:Naive Set Theory 554: 496: 494: 493: 488: 476: 474: 473: 468: 466: 465: 443: 441: 440: 435: 423: 421: 420: 417:{\displaystyle } 415: 392: 390: 389: 384: 362: 360: 359: 354: 340: 338: 337: 332: 327: 326: 311: 310: 302: 296: 295: 290: 268: 266: 265: 260: 255: 254: 236: 235: 217: 216: 194: 192: 191: 186: 181: 180: 169: 160: 156: 155: 144: 135: 131: 130: 115: 114: 106: 96: 94: 93: 88: 83: 82: 77: 55: 53: 52: 47: 732: 731: 727: 726: 725: 723: 722: 721: 702: 701: 700: 699: 642: 640: 639: 632: 616: 612: 605: 589: 585: 578: 555: 551: 546: 504: 482: 479: 478: 461: 457: 449: 446: 445: 429: 426: 425: 403: 400: 399: 375: 372: 371: 348: 345: 344: 322: 318: 301: 300: 291: 277: 276: 274: 271: 270: 250: 246: 231: 227: 212: 208: 203: 200: 199: 176: 172: 151: 147: 126: 122: 105: 104: 102: 99: 98: 78: 64: 63: 61: 58: 57: 41: 38: 37: 12: 11: 5: 730: 720: 719: 714: 698: 697: 690: 683: 675: 672: 671: 654: 638: 637: 630: 610: 603: 583: 576: 548: 547: 545: 542: 541: 540: 534: 528: 522: 517: 511: 503: 500: 499: 498: 486: 464: 460: 456: 453: 433: 413: 410: 407: 382: 379: 352: 341: 330: 325: 321: 317: 314: 308: 305: 299: 294: 289: 286: 283: 280: 258: 253: 249: 245: 242: 239: 234: 230: 226: 223: 220: 215: 211: 207: 184: 179: 175: 168: 165: 159: 154: 150: 143: 140: 134: 129: 125: 121: 118: 112: 109: 86: 81: 76: 73: 70: 67: 45: 9: 6: 4: 3: 2: 729: 718: 715: 713: 710: 709: 707: 696: 691: 689: 684: 682: 677: 676: 670: 668: 664: 660: 655: 652: 648: 647: 643: 633: 631:9783540440857 627: 623: 622: 614: 606: 604:9780387943695 600: 596: 595: 587: 579: 577:9780387900926 573: 569: 565: 564: 559: 558:Halmos, P. R. 553: 549: 538: 535: 532: 529: 526: 523: 521: 518: 515: 512: 509: 506: 505: 484: 462: 454: 431: 408: 396: 380: 377: 370: 366: 350: 342: 328: 323: 319: 315: 303: 292: 269:to the value 251: 247: 243: 240: 237: 232: 228: 224: 221: 218: 213: 209: 198: 177: 173: 166: 163: 157: 152: 148: 141: 138: 132: 127: 123: 116: 107: 84: 79: 43: 35: 34:set-theoretic 31: 30: 29: 27: 23: 19: 667:expanding it 656: 641: 620: 613: 593: 586: 562: 552: 21: 15: 706:Categories 659:set theory 544:References 395:surjection 22:projection 18:set theory 497:explicit. 307:→ 244:× 241:⋯ 238:× 225:× 222:⋯ 219:× 164:… 139:… 111:→ 26:functions 560:(1960), 502:See also 363:to its 195:of the 628:  601:  574:  170:  161:  145:  136:  657:This 424:when 663:stub 626:ISBN 599:ISBN 572:ISBN 20:, a 16:In 708:: 566:, 32:A 694:e 687:t 680:v 669:. 635:. 608:. 581:. 485:E 463:E 459:] 455:x 452:[ 432:E 412:] 409:x 406:[ 381:, 378:E 351:x 329:. 324:j 320:x 316:= 313:) 304:x 298:( 293:j 288:j 285:o 282:r 279:p 257:) 252:k 248:X 233:j 229:X 214:1 210:X 206:( 183:) 178:k 174:x 167:, 158:, 153:j 149:x 142:, 133:, 128:1 124:x 120:( 117:= 108:x 85:, 80:j 75:j 72:o 69:r 66:p 44:j

Index

set theory
functions
set-theoretic
Cartesian product
equivalence class
equivalence relation
surjection
Cartesian product
Projection (mathematics)
Projection (measure theory)
Projection (linear algebra)
Projection (relational algebra)
Relation (mathematics)
Halmos, P. R.
Naive Set Theory
Undergraduate Texts in Mathematics
ISBN
9780387900926
An Introduction to Analysis
ISBN
9780387943695
Set Theory: The Third Millennium Edition
ISBN
9783540440857
Stub icon
set theory
stub
expanding it
v
t

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.