Discrete Mathematics Answers

(b) Find f ◦ g and g ◦ f, where f(x) = x
2 + 1 and g(x) = x + 2, are functions from R to
R.
(c) Find f + g and fg for the functions f and g given in part (b).

Find f ◦ g and g ◦ f, where f(x) = x
2 + 1 and g(x) = x + 2, are functions from R to
R

Let f : Z → Z be such that f(x) = x + 1. Is f invertible, and if it is, what is it’s
inverse?

) (i) Let f be the function from {a, b, c} to {1, 2, 3} such that f(a) = 2, f(b) = 3 and
f(c) = 1. Is f invertible, and if it is, what is it’s inverse?

) Determine whether each of these functions is a bijection from R to R.
(i) f(x) = -3x + 4
(ii) f(x) = 2x + 1
(iii) f(x) = x
2 + 1
(iv) f(x) = -3x
2 + 7
(v) f(x) = (x+1)
(x+2)

(a) Give an explicit formula for a function from the set of integers to the set of positive integers that is
(i) one-to-one, but not onto
(ii) onto, but not one-to-one
(iii) one-to-one and onto
(iv) neither one-to-one nor onto

(i) Determine whether each of these functions from [a,b,c,d] to itself is one-to-one.
(a) f(a) = b, f(b) = a, f(c) = c, f(d) =d
(b) f(a) = b, f(b) = b, f(c) = d, f(d) = c
(c) f(a) = d, f(b) = b, f(c) = c, f(d) = d
(ii) Which functions in part (i) are onto?

Let H(x) be the statement “x is hardworking”, N(x) be the statement “x is naughty” and C(x) be the
statement “ x is clever”, where the domain for x consists of all students. Use quantifications to
express each of the following statements.
a) Some students are clever but naughty.
b) Not all students are clever and hardworking.
c) Some students are clever, hardworking and not naughty.
d) All students are clever, or hardworking or naughty.

Let P(x) be the statement “x likes subject mathematics,” where the domain for x consists of all
students. Express each of these quantifications in English.
a) ∃