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GHANA TECHNOLOGY UNIVERSITY COLLEGE (Faculty of Computing and Information Systems) MID- SEMESTER EXAMINATIONS, APRIL 2021 MATH 171: DISCRETE STRUCTURES Answer all questions

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GHANA TECHNOLOGY UNIVERSITY COLLEGE

(Faculty of Computing and Information Systems)

MID- SEMESTER EXAMINATIONS, APRIL 2021

MATH 171: DISCRETE STRUCTURES

Answer all questions.

1. a. Define the following with and example;

i. a function

ii . one- to- one function

b. The functions g and h are defined by g: x ® x² — 4 and f: X®x/z-x. Where x is a real number.

i. State the domain of each of the two functions

ii. Determine whether or not f is a one -to- one function

iii. Find the inverse of the function f

c. The functions h and m are defined on the set R of real numbers by h: x ® ax + b and m: x ® cx -d, where a, b, c and d are constants. If h o m = m o h, show that (c-1)b = (a-1)d

2. Let R be the relation on the set A = {1, 2, 3, 4, 5, 6, 7} defined by the rule (a,b) Î R if the integer (a — b) is divisible by 4. List the elements of R and its inverse?

b) Check whether the relation R on the set S = {1, 2, 3} is an equivalent relation where

R = {(1,1), (2,2), (3,3), (2,1), (1,2), (2,3), (1,3), (3,1)}. Which of the following properties R has: reflexive, symmetric, anti-symmetric, transitive? Justify your answer in each case?

c) Let S = {a, b, c}and R = {(a,a), (b,b), (c,c), (b,c), (c, b)}, find [al], [b] and [c] (that is the equivalent class of a, b, and c). Hence or otherwise find the set of equivalent class of a,b and c?

3. a. Define the following with an example;

1. paths

ll. simple graph

b. Draw the graph with the adjacency matrix

0	3	0	2
3	0	1	1
0	1	1	2
2	1	2	0

 

with respect to the

ordering of vertices, a, b, c, d.

i. Find the degree of each vertex in your graph from part (a) above.

ll. How many walks of length 2 are there from the vertex c to c? How many of these walks are paths?

4. a. Define the following Terms giving one example each:

1. Partial Ordering Relations

li. Equivalence relations

b. Answer these questions for the partial order represented by the following Hasse diagram.

1. Find the maximal elements.

11. Find the minimal elements.

il. Is there a greatest element?

iv. Is there a least element?

v. Find all upper bounds of {m, k, s}.

vi. Find all lower bounds of {c, d, t}.

vil. Find the greatest lower bound of {u, k, m} if it exists.

Vill. Find the least upper bound of {b, k, t} if it exists.