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

3

## Equivalence Relations

### Reflexive, Symmetric, Transitive

4

## Reflexive

### For every x in A, xRx

5

## Symmetric

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

6

## 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

8

## 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

9

## 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

10

## 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)

12

## 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}

16

## Inverse image F(Y) is

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

17

## f(C n D) subset of

### f(C) n f(D)

18

## f(C u D) equals

### f(C) u f(D)

19

## inverse f( E n F) equals

### inverse f(E) n inverse f(F)

20

## inverse f(E u F) equals

### inverse f(E) u inverse f(F)

21

## a set S is countable if and only if

### S is finite or denumerable

22

## 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