College Algebra 9th txtbk

6 CHAPTER R BASIC ALGEBRAIC OPERATIONS
Z BASIC PROPERTIES OF THE SET OF REAL NUMBERS
Let R be the set of real numbers, and let x, y, and z be arbitrary elements of R.
Addition Properties
Closure: x y is a unique element in R.
Associative: (x y) z x (y z)
Commutative: x y y x
Identity: 0 is the additive identity; that is, 0 x x 0 x for all
x in R, and 0 is the only element in R with this property.
Inverse: For each x in R, x is its unique additive inverse; that is,
x (x) (x) x 0, and x is the only element in R
relative to x with this property.
Multiplication Properties
Closure: xy is a unique element in R.
Associative: (xy)z x( yz)
Commutative: xy yx
Identity: 1 is the multiplicative identity; that is, for all x in R,
(1)x x(1) x, and 1 is the only element in R with this
property.
Inverse: For each x in R, x 0, x 1 is its unique multiplicative
inverse; that is, xx 1 x 1 x 1, and x 1 is the only element
in R relative to x with this property.
Combined Property
Distributive:
x(y z) xy xz
(x y)z xz yz
EXAMPLE 2 Using Real Number Properties
Which real number property justifies the indicated statement?
(A) (7x)y 7(xy)
(B) a(b c) (b c)a
(C) (2x 3y) 5y 2x (3y 5y)
(D) (x y)(a b) (x y)a (x y)b
(E) If a b 0, then b a.
SOLUTIONS
(A) Associative ()
(B) Commutative ()
(C) Associative ()
(D) Distributive
(E) Inverse ()