Suppose that A and B are non-empty sets, and f: A → B is a function. (a) Prove that fis injective if and only if there exists a function g: B → A such that g(f(x)) = x for all A. (b) Prove that fis surjective if and only if there exist a function h: B→ A such that f(h(y)) = y for all y = B. (c) Prove that if f is both injective and surjective, then there exists a unique function g: B→ A such that g(f(x)) : x for all x A and f(g(y)) y for all y = B.

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?

Determine if the statemment is true or false. If the statement is false, then correct it and make it true. If the function f increases on the interval -,x1 and decreases on the interval x1,, then fx1 is a local minimum value.

A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.