For each statement below decide whether it is true or false

For each statement below, decide whether it is true or false. Then prove it or give a counterexample.

(6) A composition of injective mappings is injective.

(7) A function X f ? Y is bijective if and only if it is both injective and surjective.

(6) A composition of injective mappings is injective. (7) A function X-Y is bijective if and only if it is both injective and surjective.

Solution

A function maps elements from its domain to elements in its co domain.

The function is injective (one-to-one) if every element of the co domain is mapped to by at most one element of the domain.

The function is surjective (onto) if every element of the co domain is mapped to by at   least one element of the domain.

The function is bijective (one-to-one and onto or one-to-one correspondence) if every element of the co domain is mapped to by exactly one element of the domain. (That is, the function is both injective and surjective.)

Thus, by definition itself, the statement number 7 is true.

The statement number 6 is also true. It is proved as under:

Let us consider injective functions f : A B and g : B C. We want to show that g f is also injective. We do this by proving that if (g f)(a1) = (g f)(a2), then a1 = a2. If (gf)(a1) = (gf)(a2), then g(f(a1)) = g(f(a2)). Since g is an injective function, this implies that f(a1) = f(a2). Since f is also an injective function, this implies that a1 = a2. Thus, (g f)(a1) = (g f)(a2) a1 = a2, so g f is injective.