Definitions

Question 1

definition

Answer 1

An exact statement of the meaning of a word

Question 2

theorem

Answer 2

A proved mathematical statement

Question 3

lemma

Answer 3

A proved result, typically less important than a theorem; often used to prove theorems.

Question 4

proposition

Answer 4

A proved statement, typically less important than a lemma

Question 5

corollary

Answer 5

The direct result of a theorem, lemma, or proposition

Question 6

axiom

Answer 6

A statement assumed to be true.

Question 7

set

Answer 7

A well-defined collection of objects

Question 8

ℕ

Answer 8

{0, 1, 2, 3, …}

Question 9

ℤ

Answer 9

{…, -2, -1, 0, 1, 2, …}

Question 10

ℤ+

Answer 10

{ 1, 2, 3, 4, 5, …}

Question 11

ℚ

Answer 11

{ a/b | a,b∈ℤ, b ≠ 0}

Question 12

ℚ’ / ℍ

Answer 12

{ x | x is a decimal number that neither terminates or repeats}

Question 13

ℝ

Answer 13

{ x | x ∈ ℚ⋃ℚ’ }

Question 14

ℂ

Answer 14

{ a + bi | a,b ∈ ℝ, i^2 = -1 }

Question 15

The empty set

Answer 15

The set that contains no elements

Question 16

A⋃B

Answer 16

{ x | x∈A, x∈B, or both }

Question 17

A⋂B

Answer 17

{ x | x∈A and x∈B }

Question 18

Ᾱ / A^c

Answer 18

{ x | x∉A }

Question 19

A - B

Answer 19

{ x∈A | x∉B }

Question 20

A⊆B

Answer 20

∀x, if x∈A, x∈B.

Question 21

A⊂B

Answer 21

∀x, if x∈A, x∈B,

AND ∃ y∈B ∋ y∉A.

Question 22

power set

Answer 22

Let A be a set.

Then the power set of A is the set containing all subsets of A; denoted P(A).

Question 23

AxB

Answer 23

Let A,B be sets.

Then AxB = { (a,b) | a∈A, b∈B }

Question 24

DeMorgan’s Laws

Answer 24

~(P and Q) = ~P or ~Q

~(P or Q) = ~P and ~Q

Question 25

partition

Answer 25

Let X be a set.
A partition S of X is a collection of sets such that:

1. If A,B∈S, then A⋂B = ∅, unless A=B, and
2. ⋃(A∈S) A = X

Question 26

statement

Answer 26

A sentence that is either true or false, and can be evaluated.

Question 27

open statement

Answer 27

A statement with a variable involved

Question 28

WFF

Answer 28

A well-formed formula.

A statement consisting of at least one statement and one connective.

Question 29

equivalent

Answer 29

Two WFF are equivalent if they have the same truth values.

Question 30

tautology

Answer 30

A statement that is always true.

Question 31

contradiction

Answer 31

A statement that is always false.

Question 32

∀

Answer 32

The universal quantifier

Question 33

∃

Answer 33

The existential quantifier

Question 34

Odd integer

Answer 34

A number n is odd if ∃ m∈ℤ ∋ n = 2m+1

Question 35

Even integer

Answer 35

A number n is even if ∃ m∈ℤ ∋ n = 2m

Question 36

Divides

Answer 36

Let a,b∈ℤ.

Then a | b if ∃ k∈ℤ ∋ ak = b

Question 37

|a|

Answer 37

= a if a ≥ 0

-a otherwise

Question 38

a ≡ b mod c

Answer 38

Let a,b,c ∈ℤ.
Then a ≡ b mod c , if c | (b-a)

e.g.
33 ≡ 3 mod 10 –> 103 R3
33 ≡ 13 mod 10 –> 102 R13
33 ≡ -7 mod 10 –> 10*4 R-7

Question 39

a ≤ b

Answer 39

a ≤ b if ∃ c ≥ 0 ∋ a+c=b

Question 40

a

Answer 40

a 0 ∋ a+c=b

Question 41

prime number

Answer 41

A number p is prime if it has exactly 2 positive divisors

p∈ℤ+

Question 42

composite number

Answer 42

Any number n∈ℤ+ (other than 1) which is not prime.

Question 43

f’(x)

Answer 43

f’(x) = lim h→0 [ f(x+h) - f(x) / h ]

…as long as the limit exists!

Question 44

local max

Answer 44

A function f has a local max at c if f(c) ≥ f(x) for ∀x close to c.

Question 45

EVT (extreme value theorem)

Answer 45

Let f be continuous on [a,b]. Then f has an absolute max at some c∈[a,b].

Question 46

Fermat theorem

Answer 46

Let f be continuous function and have a local max at c. If f’(c) exists, then f’(c) = 0.

Question 47

Rolle’s Theorem

Answer 47

Let f be a continuous function satisfying

1. f is cts on [a,b]
2. f is differentiable on (a,b)
3. f(a) = f(b).

Then ∃ c∈(a,b) ∋ f’(c) = 0.

(basically, if a function has two equal values at two distinct points, then there must be a stationary point between them, i.e. the slope of the tangent is 0)

Question 48

Triangle inequality

Answer 48

a + b | ≤ | a | + | b |

Question 49

If A is a set, what is |A|?

Answer 49

It is the number of elements in A, either finite or infinite, called the cardinality of A

Question 50

If P(A) is the power set of A, what is the cardinality of P(A)?

Answer 50

|P(A)| = 2^|A|

Question 51

What are the five basic connectives?

Let S and T be statement variables

Answer 51

1. Negation of S (~S)
2. Conjunction (S∧T)
3. Disjunction (S∨T)
4. Implication (S => T)
5. Biconditional (S T)

Question 52

What does biconditional mean?

Let S and T be statement variables

Answer 52

S T
S iff T
They imply each other, that is
S => T and T => S

Question 53

What is a mathematical proof?

Answer 53

A formal series of statements showing that if a hypothesis is true, then necessarily ‘this’ conclusion must also be true