Proofs 2 Flashcards

(24 cards)

1
Q

Domain from A to B

A

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

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2
Q

Range from A to B

A

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

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3
Q

Equivalence Relations

A

Reflexive, Symmetric, Transitive

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4
Q

Reflexive

A

For every x in A, xRx

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5
Q

Symmetric

A

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

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6
Q

Transitive

A

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

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7
Q

x congruent to y (mod) iff

A

m divides x-y

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8
Q

Partition Definition

A

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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9
Q

Anti-Symmetric

A

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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10
Q

Well Ordering Principle

A

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

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11
Q

If a within D, then

A

f(a) within f(D)

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12
Q

If a within inverse f(E)

A

then f(a) within E

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13
Q

If f(a) within E

A

then a within inverse f(E)

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14
Q

If f(a) within f(D)

A

then a within D

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15
Q

F(X) = image of set X is

A

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

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16
Q

Inverse image F(Y) is

A

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

17
Q

f(C n D) subset of

18
Q

f(C u D) equals

19
Q

inverse f( E n F) equals

A

inverse f(E) n inverse f(F)

20
Q

inverse f(E u F) equals

A

inverse f(E) u inverse f(F)

21
Q

a set S is countable if and only if

A

S is finite or denumerable

22
Q

S is finite or denumerable

A

set S is countable

23
Q

Denumerable

A

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

24
Q

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