Discrete Mathematics Answers

(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

F(x)=x is one to one function true or false

13 and 65 are relatively prime? True or false

Q1 Find the floor and ceiling Function of x and . Take any value of x and y. (5)

X Floor Ceiling
-X.Y (x point y)
X / Y (x divided by y)

Q4 Write down any 4*4 matrix having only zeros and ones. (3*5=15)

a) Draw the Directed graph of that matrix.
b) List the ordered pairs in the relation on set {1, 2, 3, 4} corresponding to this matrix.
c) Determine whether the relations on this graph/ matrix are Reflexive, Symmetric and Anti-symmetric.
d) Determine whether the relation for this graph is equivalence or not?