Chapter 7

Question 1

commutativity of addition for the real #’s [axiom]

Answer 1

For all x, y ∈ R, we have

x + y = y + x

Question 2

associativity of addition for the real #’s [axiom]

Answer 2

For all x, y, z ∈ R, we have

(x + y) + z = x + (y + z)

Question 3

distributivity for the real #’s [axiom]

Answer 3

For all x, y, z ∈ R, we have

x ∙ (y + z) = x ∙ y + x ∙ z

Question 4

commutativity of multiplication for the real #’s [axiom]

Answer 4

For all x, y ∈ R, we have

x ∙ y = y ∙ x

Question 5

associativity of multiplication for the real #’s [axiom]

Answer 5

For all x, y, z ∈ R, we have

(x ∙ y) ∙ z = x ∙ (y ∙ z)

Question 6

additive identity for the real #’s [axiom]

Answer 6

There exists a real number 0 satisfying

∀ x ∈ R, x + 0 = x

This element 0 is called an additive identity, or an identity element for addition.

Question 7

multiplicative identity for the real #’s [axiom]

Answer 7

There exists a real number 1 such that

1 ≠ 0 and ∀x ∈ R, x · 1 = x

The element 1 is called a multiplicative identity, or an identity element for multiplication.

Question 8

additive inverse for the real #’s [axiom]

Answer 8

For each x ∈ R, there exists a real number, denoted −x, such that

x + (−x) = 0

The element −x is called an additive inverse of x.

Question 9

multiplicative inverse for the real #’s [axiom]

Answer 9

For each x ∈ R − {0}, there exists a real number,
denoted^-1, such that

x · x^-1 = 1

The element x −1 is called a multiplicative inverse of x

Question 10

definition of subtraction for the real #’s [definition]

Answer 10

x − y := x + (−y), x, y ∈ R

Question 11

definition of division for the real #’s [definition]

Answer 11

If x, y ∈ R and x ≠ 0, we define

y/x = yx^-1

Question 12

the division function for the real #’s [function]

Answer 12

R × (R - {0}) → R

Question 13

1/x can also be expressed as… [note]

Answer 13

x^-1