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620 Appendix B 11 Sets and Functions

Definition B.2.2 Two functions f : A ~ B and g : A' ~ B' are said to be

equal if A= A', B = B', and 'Vx EA, f(x) = g(x).

Definition B.2.3 A function f: A~ Bis one-to-one (or 1-1) if 'Va, a' EA,

j(a) = j(a') =}a= a'. Equivalently, a =f:. a'=} f(a) =f:. f(a'). That is, a function

is 1-1 iff^6 different inputs always result in different outputs.

Definition B.2.4 A function f : A ~ B is onto B if R(f) = B. That is,

f : A ~ B is onto B iff every element of B is an output of f.

Example B.2.5 The function f: JR~ JR given by f(x) = x^2 is

(a) not 1-1, since f (1) = 1 and f (-1) = 1;

(b) not onto JR, since R(f) = [O, +oo) =f:. R 0

Example B.2.6 The function f: JR~ JR given by f(x) = x^3 is

(a) 1-1, since

f(a) = j(b) =} a^3 = b^3

=} a^3 - b^3 = 0

=} (a - b) (a^2 + ab + b^2 ) = 0

=} a - b = 0, since a^2 +ab+ b^2 =f:. 0 (if b =f:. 0)

=}a= b.

Note that the reason a^2 +ab+ b^2 =f:. 0 when b =f:. 0 lies in the fact that a

and bare real numbers, and the discriminant of the quadratic function g(a) =

a^2 +ab+ b^2 is D = b^2 - 4(1)(b2) = -3b^2 < 0.

(b) onto JR, since 'Vx E JR, 3 ..yi E JR and f( ..yi) = x, so x E R(f). 0

Definition B.2.7 A function f: A~ B that is both 1-1 and onto is said to

b e a 1-1 correspondence.

Definition B.2.8 Two sets A and B are said to have the same cardinal

number (of elements) if 31-1 correspondence f : A~ B.

IMAGES AND INVERSE IMAGES OF SETS

Definition B.2.9 Suppose f: A~ Bis a function and C ~ A and D ~ B.

Then

f(C) = {f(x) : x EC} ;

1-^1 (D) = {x: f(x) ED}.

6. For the definition of "iff" see Definition A.1.9.