Discrete Mathematics Answers

(iv) f : N → Y such that f(n) = 2n + 1,
where Y = {y ∈ N : y = 4n + 3 for some n ∈ N}

(b) Determine whether each of following functions are invertible. If it is an invertible, find
it’s inverse.
(i) f : R → R such that f(x) = 3x + 10
(ii) f : R → R such that f(x) = x
2
- 1

(a) Let A = {a,b,c,d,e,f} and B = {1,2,3,4,5,6}. Determine whether each of following functions from A to B are invertible. If it is an invertible, find it’s inverse.
(i) f1 = {(a,1), (b,2), (c,3), (d,4), (e,5), (f,6)}
(ii) f2 = {(a,2), (b,5), (d,2), (c,3), (e,5), (f,6)}
(iii) f3 = {(b,3), (d,6), (a,1), (c,3), (e,4), (f,5)}
(iv) f4 = {(a,6), (b,5), (c,1), (f,2), (d,4), (e,3)}

(b) Determine whether each of these function is a bijection
(i) f : N → N such that f(n) = n
2
(ii) f : N → N such that f(n) = n + 3
1
(iii) f : R → R such that f(x) = x
3
(iv) f : R → R such that f(x) = x
2 + 1
(v) f : N → N such that
f =
(
n − 1 : n is odd
n + 1 : n is even

Let g : R → R defined by the equation g(x) = x
2 + x. Let H ⊆ R and
H = {y ∈ R : 6 ≤ y ≤ 12}. Then determine the inverse image g
−1
(H).

(a) (i) Let f : R → R be defined by the equation f(x) x
2 + 1. Let H ⊆ R
and H = { y ∈ R : 5 ≤ y ≤ 10}. Then determine the inverse image f
−1
(H).

Let A = {a,b,c,d} and B = {c,d,e,f,g}.
Let R1 = {(a,c), (b,d), (c,e)}
R2 = {(a,c), (a,g), (b,d), (c,e), (d,f)}
R3 = {(a,c), (b,d), (c,e), (d,f)}
Justify which of the given relation is a function from A to B.
(c) Let f be a real valued function defined by f(x) = 1
x2−9
.
(i) What is the domain of f?
(ii) What is the range of f?
(iii) Represent f as a set of ordered pairs.

1. (a) Let A = {1,2,3,4} and B={0,3,6,8,12,15}. Consider a rule f(x) = x
2 − 1, x ∈ A. Is f be a
function from A to B?

Find formula for tha sequences with tha following first five terms 1,1/2,1/4,1/8,1/16

a) 04 points Draw Venn diagram to describe sets A, B, and C that satisfy the given conditions. AC B, CC B, AnC +0. (b) 03 points For each integer m, let Tm = {m2, m*}. How many elements are in each of T-3, T-1, To, T1? Give justification. (c) 03 points Let A = {-2,0, 2}, B = {4,6, 8} and define a relation T from A to B as follows: For all (r, y) E A x B, (r, y) € T means that is an integer. Write T as a set of ordered pairs and find the %3! %3D %3D domain and co-domain of T