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

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finite-card.1

February 21, 2014

Mathematics for Computer Science

MIT 6.042J/18.062J

Finite Cardinality

Albert R Meyer

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February 21, 2014

Mapping Rule (bij)

A bijection from A to B implies

|A| = |B| for finite A, B

Albert R Meyer

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size of the power set

# subsets of a finite set A? |pow(A)| ?

for A = {a, b, c}, pow(A) = {

Albert R Meyer

, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c} }

finite-card.4

February 21, 2014

pow(A) bijection to bit-strings

A : {a0, a1, a2, a3, a4, … , an-1}

subset: {a0, a2, a3, … , an-1}

string: 1 0 1 1 0 … 1 this defines a bijection, so

# n-bit strings = |pow(A)|

Albert R Meyer

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February 21, 2014

pow(A) bijection to bit-strings

every computer scientist knows #n-bit strings, so

Corollary:

|pow(A)| = 2n

|A|

Albert R Meyer

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function & surjective

≥ 1 arrow in

Albert R Meyer

A B

≤ 1 arrow out

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February 21, 2014

Mapping Rule (surj)

function: A B

Albert R Meyer

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February 21, 2014

Mapping Rule (surj)

[≤ 1 out] : A

B

IMPLIES |A| ≥ #arrows.

surjection: A B

Albert R Meyer

→ → →

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finite-card.9

February 21, 2014 Albert R Meyer

Mapping Rule (surj)

[≤ 1 o t] : u

A B

IMPLIES |A| ≥ #arrows.

:

[≥1 in] A

B

IMPLIES #arrows ≥|B|.

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February 21, 2014 Albert R Meyer

Mapping Rule (surj)

Surjective function from A to B implies

|A| ≥ |B| for finite A, B

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February 21, 2014 Albert R Meyer

A B

injection archery

≤ 1 arrow in

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Mapping Rule (inj)

total [≥1 out] IMPLIES |A| ≤ #arrows injection [≤1 in] IMPLIES #arrows ≤ |B|

Albert R Meyer

→ →

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Mapping Rule (inj)

Total injective relation from A to B implies

|A| ≤ |B| for finite A, B

Albert R Meyer February 21, 2014

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February 21, 2014 Albert R Meyer

“jection” relations

A bij B ::= ∃ bijection:A B A surj B ::= ∃ surj func:A B A inj B ::= ∃ total in j relation:A B

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February 21, 2014 Albert R Meyer

Mapping Lemma

A bij B IFF A = B A surj B IFF A ≥ B

A inj B IFF A ≤ B

for finite A, B

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February 21, 2014 Albert R Meyer

Familiar “size” properties

|A| =|B|= | C| IMPLIES |A|=|C| |A|≥ |B|≥ | C| IMPLIES |A|≥|C| |A| ≥ |B| ≥ |A| IMPLIES |A|=|B|

for finite A, B, C

→ → →

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Familiar “size” properties

A bij B bij C IMPLIES A bij C A surj B surj C IMPLIES A surj C A surj B surj A IMPLIES A bij B

for finite A, B, C

by the Mapping Lemma

Albert R Meyer February 21, 2014

finite-card.17

Familiar “size” properties

A bij B bij C IMPLIES A bij C A surj B surj C IMPLIES A surj C A surj B surj A IMPLIES A bij B

for infinite A, B, C, also 1st two implications: easy 3rd is tricky

Albert R Meyer February 21, 2014

finite-card.21

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6.042J / 18.062J Mathematics for Computer Science

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