Chapter 1

Question 1

commutativity of addition [axiom]

Answer 1

a + b = b + a

Question 2

associativity of addition [axiom]

Answer 2

(a + b) + c = a + (b + c)

Question 3

distributivity [axiom]

Answer 3

a * (b + c) = a * b + a * c

Question 4

commutativity of multiplication [axiom]

Answer 4

a * b = b * a

Question 5

associativity of multiplication [axiom]

Answer 5

(a * b) * c = a * (b * c)

Question 6

additive identity [axiom]

Answer 6

There exists an interger 0 such that
a + 0 = a
for all a E Z.

The element 0 is called an additive identity.

Question 7

multiplicative identity [axiom]

Answer 7

There exists an interger such that
1 =/= 0 and a * 1 = a
for all aEZ.

The element 1 is called a multiplicative identity.

Question 8

additive inverse [axiom]

Answer 8

For each a E Z, there exists an interger, denoted -a, such that
a + (-a) = 0 .

The element -a is called the additive inverse of a.

Question 9

cancellation property [axiom]

Answer 9

If a, b, c E Z,
a * b = a * c,
and a =/= 0,
then b = c .

Question 10

reflexivity of (in)equality [axiom]

Answer 10

a = a

Does not apply to =/= .

Question 11

symmetry of (in)equality [axiom]

Answer 11

If a = b, then b = a .

Also applies to =/= .

Question 12

transitivity of (in)equality [axiom]

Answer 12

If a = b and b = c, then a = c .

Does not apply to =/= .

Question 13

replacement property [axiom]

Answer 13

If a, b, c E Z and a = b,

then a + c = b + c .

Question 14

uniqueness of the additive inverse [proposition]

Answer 14

If a, b E Z,
and a + b = 0 ,
then b = -a

Question 15

uniqueness of the additive identity [proposition]

Answer 15

If a E Z has the property that
b + a = b for all b E Z,
then a = 0 .

Question 16

uniqueness of the multiplicative identity [proposition]

Answer 16

If a E Z has the property that, for some nonzero b E Z,
b * a = b,
then a = 1 .