Discrete Mathematics Answers

For the relation R = {(p,p) ,(q,p),(q,q),(r,r),(r,s),(s,s) ,(s,m) ,(m,m)}

1.Using warshall algorithm find the transitive closure R* of R

2.write matrix representation of R*

3.Check whether the relation of R* is an equivalence relation or a partial order.

range of the function f(x) = 5x − 3 defined by the domain {0,1,2}

For each of the following sets,

a) S={1,2,3}, T ={a, b, c}

b) S={a, b}, T ={1,2,3,4}

c) S={1,2,3,4}, T ={a, b}

Determine whether

1. There is a one-to-one function f:S→T;

2. There is an onto function f:S→T; and

3. If there is a bijective function f:S→T.

4. For each (1–3) if such a function exists, explicitly give it. If no function exists give a short explanation?

. What is the value of x after each of these statements is encountered in a computer program, if x = 1 before the statement is reached? a) if x + 2 = 3 then x := x + 1

. What is the value of x after each of these statements is encountered in a computer program, if x = 1 before the statement is reached? a) if x + 2 = 3 then x := x + 1

1. Find these terms of the sequence {an}, where an =

2 · (−3)n + 5n.

a) a0

b) a1

c) a4

d) a5

In a school, 100 students have access to three software packages A, B and C.

28 did not use any software

8 used only packages A

26 used only packages B

7 used only packages C

10 used all three packages

13 used both A and B

If twice as many students used package B as package A, Using Venn Diagram:

1. Calculate the no. of students using packages A & C only but not B.

2. Calculate the no. of students using packages C & B only but not A.

From a total of 125 employees, 102 employees are assigned to perform ONLY ONE programming job, either system programming or application programming. And 10 employees are not assigned to perform any of those programming jobs. Overall, 52 employees are assigned to perform ONLY system programming jobs.

a. How many employees are assigned to perform both system programming jobs and application programming jobs?

b. How many employees are assigned to perform system programming jobs or application programming jobs?

c. How many employees are assigned to perform system programming jobs?

d. How many employees are NOT assigned to perform system programming jobs?

Construct a K-map for F(x, y, z) = xz + yz + xyz. Use this K-map to find the implicants, prime implicants, and essential prime implicants of F(x, y, z).

Proof that an undirected graph has an even number of vertices of odd degree.