Test 2 Theorem List Flashcards Preview

TXST: Intro To Advanced Math > Test 2 Theorem List > Flashcards

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

If S has a LUB q and there exists a number p < q,

Then there exists an element in S such that p < x <= q