Mathematics for Computer Science

4.5. Finite Cardinality 105

4. all three parts ofR.

Properties:

(a)surjective

(b)injective

(c)total

(d)function

(e)bijection

Problem 4.18.
For each of the following real-valued functions on the real numbers, indicate whether
it is a bijection, a surjection but not a bijection, an injection but not a bijection, or
neither an injection nor a surjection.

(a)x!xC 2

(b)x!2x

(c)x!x^2

(d)x!x^3

(e)x!sinx

(f)x!xsinx

(g)x!ex

Problem 4.19.
Letf WA!BandgWB!Cbe functions andhWA!Cbe their composition,
namely,h.a/WWDg.f.a//for alla 2 A.
(a)Prove that iffandgare surjections, then so ish.

(b)Prove that iffandgare bijections, then so ish.

(c)Iff is a bijection, then so isf^1.