Flashcards in Test 2 Theorem List Deck (38):

1

## Definition of absolute value

### x if x >= 0 and -x if x < 0

2

## |x| =

### |-x|

3

## |x| >=

### x

4

## |x||y| =

### |xy|

5

## |x/y| when y = 0

### |x|/|y|

6

## |x + y| <=

### |x| + |y|

7

## |x| - |y| <=

### |x - y|

8

##
y - c < x < y + c

Is equivalent to

### |x - y| < c

9

##
For each positive number c,

|A - B| < c is equivalent to

### A = B

10

## If |A - B| < d/2 and |B - C| < d/2

### |A - C| < d

11

## If |x - p| < d,

### |x| < d + |p| and |p| - d < |x|

12

## Definition of the set of counting numbers

### n is in the set iff n - 1 is in the set or n = 1

13

## n is an integer iff

### Either n is in the set of all positive integers, -n is in the set of all positive integers, or n = 0

14

## a divides b (a|b)

### b = ac

15

## a is equivalent to bmod(n)

### a = b + qn (q is an integer)

16

## If a is equivalent to bmod(n) and c is equivalent to dmod(n),

### (a + c) is equivalent to (b + d)mod(n) and ac is equivalent to bdmod(n)

17

## a is equivalent to bmod(n) iff

### n|(b - a)

18

## Definition of even

### n = 2q (q is an integer)

19

## Definition of odd

### n = 2q + 1 (q is an integer)

20

## If a and b are positive integers and a|b,

### a <= b

21

## Well-Ordering Principle

### If S is a subset of positive integers, then S has a least element.

22

##
If S is a subset of positive integers and:

1) 1 is in S

2) if n is in S, then n+1 is in S

### Then S contains all positive integers

23

## Induction Proof

###
1) Show a base case

2) assume n

3) show n+1

24

## Definition of Bounded Above

### A set is bounded above if there exists a number m such that all elements in a set are <= m

25

## Definition of Bounded Below

### A set is bounded below if there exists a number m such that all elements in a set are >= m.

26

## Definition of Bounded

### A set is bounded iff it Bounded both above and below

27

## If S and T are bounded,

### Then S U T is bounded

28

## If S is bounded and A is a subset of S,

### Then A is bounded

29

## S is bounded iff

### There exists a number q > 0 such that |x| <= q for all elements in S

30

## If S is bounded and U is a set containing all upper bounds of S,

### Then U is bounded below

31

## Definition of Least Upper Bound

###
W is a LUB of S iff:

1) W is an upper bound of S

2) W <= all other upper bounds of S

32

## Definition of Greatest Lower Bound

###
W is a GLB of S iff:

1) W is an lower bound of S

2) W >= all other lower bounds of S

33

## If q is an element of S and q is an upper bound,

### Then q is a LUB if S

34

## If p and q are LUB if S,

### Then p = q

35

## If S is bounded above,

### Then S has a LUB

36

## If S has a LUB q and T contains all upper bounds of S,

### Then q is the GLB of T

37

## If S is bounded below,

### Then S has a GLB

38