Finite A to B implies |A| = |B| Cardinality for finite A, B - PowerPoint PPT Presentation

Mathematics for Computer Science Mapping Rule (bij) MIT 6.042J/18.062J A bijection from Finite A to B implies |A| = |B| Cardinality for finite A, B finite-card .1 finite-card .2 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 size of the power set pow(A) bijection to bit-strings # subsets of a finite set A? : { a 0 , a 1 , a 2 , a 3 , a 4 , … , a n-1 } A |pow(A)| ? subset: {a 0 , a 2 , a 3 , … , a n-1 } for A = {a, b, c}, pow(A) = string: 1 0 1 1 0 … 1 { , {a}, {b}, {c}, this defines a bijection, so {a,b}, {a,c}, {b,c}, {a,b,c} } # n-bit strings = |pow(A)| finite-card .3 finite-card .4 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 1

function & surjective pow(A) bijection to bit-strings ≤ 1 arrow out ≥ 1 arrow in every computer scientist knows #n-bit strings, so A Corollary: B |pow(A)| = 2 n |A| finite-card .5 finite-card .6 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 Mapping Rule (surj) Mapping Rule (surj) [ ≤ 1 out] : A function: A B B → → IMPLIES |A| ≥ #arrows. surjection: A B → finite-card .7 finite-card .8 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 2

Mapping Rule (surj) Mapping Rule (surj) Surjective function A B [ ≤ 1 o t] : u → from A to B implies IMPLIES |A| ≥ #arrows. |A| ≥ |B| : [ ≥ 1 in] A B → IMPLIES #arrows ≥ |B|. for finite A, B finite-card .9 finite-card .10 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 injection archery Mapping Rule (inj) ≤ 1 arrow in total [ ≥ 1 out] IMPLIES |A| ≤ #arrows A injection [ ≤ 1 in] IMPLIES B #arrows ≤ |B| finite-card .11 finite-card .12 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 3

Mapping Rule (inj) “jection” relations Total injective relation A bij B ::= ∃ bijection:A B → from A to B implies A surj B ::= ∃ surj func:A B → |A| ≤ |B| A inj B ::= ∃ total in j relation:A B → for finite A, B finite-card .13 finite-card .14 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 Mapping Lemma Familiar “size” properties |A| = |B| = | C| IMPLIES |A| = |C| A bij B IFF A = B |A| ≥ |B| ≥ | C| IMPLIES |A| ≥ |C| A surj B IFF A ≥ B |A| ≥ |B| ≥ |A| IMPLIES |A| = |B| A inj B IFF A ≤ B for finite A, B, C for finite A, B finite-card .15 finite-card .16 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 4

Familiar “size” properties Familiar “size” properties A bij B bij C IMPLIES A bij C A bij B bij C IMPLIES A bij C A surj B surj C IMPLIES A surj C A surj B surj C IMPLIES A surj C A surj B surj A IMPLIES A bij B A surj B surj A IMPLIES A bij B for infinite A, B, C, also for finite A, B, C 1 st two implications: easy by the Mapping Lemma 3 rd is tricky finite-card .17 finite-card .21 Albert R Meyer February 21, 2014 Albert R Meyer February 21, 2014 5

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