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Flashcards in Proofs 2 Deck (24):
1

Domain from A to B

Dom(R) for every x in A: there exists y in B s.t xRy

2

Range from A to B

Rng(R) for every y in B: there exists a in A s.t xRy

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Equivalence Relations

Reflexive, Symmetric, Transitive

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Reflexive

For every x in A, xRx

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Symmetric

For every x and y in A, if xRy then yRx

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Transitive

For every x,y,z in A, if xRy and yRz, then xRz

7

x congruent to y (mod) iff

m divides x-y

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Partition Definition

If x in P, then x does not equal the empty set
If x in P and y in P, then x = y or x and y = empty set
Union of x in P = A

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Anti-Symmetric

A relation R on a set A is anti-symmetric iff for every x,y in A, if xRy and yRx then x=y

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Well Ordering Principle

If A subset Naturals and A doesn't equal empty set then A contains a smallest element

11

If a within D, then

f(a) within f(D)

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If a within inverse f(E)

then f(a) within E

13

If f(a) within E

then a within inverse f(E)

14

If f(a) within f(D)

then a within D

15

F(X) = image of set X is

{ y e B: y = f(x) for some x e X}

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Inverse image F(Y) is

{ x e A: f(x) e Y}

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f(C n D) subset of

f(C) n f(D)

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f(C u D) equals

f(C) u f(D)

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inverse f( E n F) equals

inverse f(E) n inverse f(F)

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inverse f(E u F) equals

inverse f(E) u inverse f(F)

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a set S is countable if and only if

S is finite or denumerable

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S is finite or denumerable

set S is countable

23

Denumerable

Bijective / one to one correspondence to the sets of Naturals; able to be counted by bijection with the infinite sets of integers

24

Bijective / one to one correspondence to the sets of Naturals; able to be counted by bijection with the infinite sets of integers

Denumerable