Vocab

Question 1

a is congruent to b modulo n

Answer 1

Let n be a positive integer. If a and b are integers we say that a is congruent to b modulo n and write, equivalently, a ≡ b (mod n)

Question 2

Rational

Answer 2

A real number r is rational if and only if there are integers a and b, with b != 0 such that r = a /b

Question 3

Irrational

Answer 3

Any number that is not rational

Question 4

Composite

Answer 4

An integer, n, is called composite if and only if n > 1 and we can write n = rs for some integer r and s, such that 1 <r< n and 1<s< n

Question 5

Axioms for +

Answer 5

1. Associatively of Addition
2. Existence of additive identity
3. Additive inverse property
4. Commutativity of addition

Question 6

Axioms for x

Answer 6

1. Associatively of multiplication
2. Existence of multiplicative identity
3. Commutativity of multiplication
4. Distributive property

Question 7

Associatively of Addition

Answer 7

For all integer x, y, and z, it is true that (x+y) +z = x + (y+z)

Question 8

Existence of additive identity

Answer 8

There is an element 0 ∈ Z such that x+0 = x, where x is any integer

Question 9

Commutativity of addition

Answer 9

For all integers x and y, it is true that x+y = y+x.

Question 10

Additive inverse property

Answer 10

For each x ∈ Z, there exists −x such that x + (−x) = 0

Question 11

Associativity of multiplication

Answer 11

For all integers x,y, and z, it is true that x(yz) = (xy)z

Question 12

Existence of a multiplicative identity

Answer 12

There exists an element, 1!=0 in Z such that for all integers, x, x · 1 = x

Question 13

Commutativity of multiplication

Answer 13

For all integers x and y, xy = yx.

Question 14

Distributive property

Answer 14

For all integers x,y, and z, it is true that x(y + z) = xy + xz and that (x + y)z = xz + yz

Question 15

Axioms for “=”

Answer 15

1. Reflexive property
2. Substitution property

Question 16

Substitution property

Answer 16

For all integers x and y such that x = y, if P (x) is true, then P (y) is also true

Question 17

Reflexive property

Answer 17

For all integers x, it is true that x = x

Question 18

Subtraction

Answer 18

Define a new operation called subtraction, denoted by − as
x − y := x + (−y),
where x and y are integers.

Question 19

Axioms for <

Answer 19

1. The Law of Trichotomy
2. The Law of Transitivity
3. The Additive Law for <
4. The Multiplicative Law for <

Question 20

The Law of Trichotomy

Answer 20

For all integers x and y, exactly one of the following is true: x > y or x < y or
x = y.

Question 21

The Law of Transitivity

Answer 21

If x < y and y < z, then x < z, where x,y, z are integers.

Question 22

The Additive Law for <

Answer 22

If x < y then x + n < y + n, where x,y,n are integers.

Question 23

The Multiplicative Law for <

Answer 23

If 0 < n and x < y, then nx < ny, where x,y,n are integers.

Question 24

Homomorphism

Answer 24

f(ab) = f(a)*f(b)