Discrete Mathematics Answers

Let A = {1, 2, 3} and R be the following relation on A;

R = {(1, 1), (1, 2), (1, 3), (3, 1), (2, 3)}

Find the Reflexive Closure of this relation R.

Write out the indicated sets by listing their elements between braces.

a) {x ∈ R: x²= 2} x {a, c, e}

b) {x ∈ R: x²= x} x {x ∈ N: x²2= x}

c) {x ∈ R: x²= 2} x {x ∈ R: |x| = 2}

d) {n ∈ Z : 2<n<5} x {n ∈ Z}

1. For each of the following relations, decide whether it is reflexive, whether it is symmetric or not, whether it is antisymmetric or not, and whether it is transitive or not on the set {1,2,3,4}? and why?

d) {(2,2), (3, 3)}

e) {(2,2), (1, 2), (3, 3)}

4. Let R be the relation on the set {1, 2, 3, 4, 5} containing the ordered pairs (1, 1), (1, 2), (1,3), (2, 3), (2, 4), (3, 1),(3, 4), (3, 5), (4, 2), (4, 5), (5, 1), (5, 2), and (5, 4).

Find R 3 and R 4

7. Draw the directed graph that represents the relation {(a, a), (a, b), (b, c), (c, b), (c, d), (d, a), (d, b)}.

8. list the ordered pairs in the relations represented by the directed graph.

Question 1. Ali has been hired by alpha company. The company asked him to give demonstration of RSA algorithm to the developers where they further use in the application. Using your discrete mathematics number theory demonstrate the following.

 Hint (you can take two prime numbers as 19 and 23 and M/P is 20). 

a.    Calculate the public key (e,n) and private key (d,n)

b.    Encrypt the message using (e,n) to get C

c.    Apply (d,n) to obtain M/P back.

d.    Write a Java program to demonstrate the keys you generated in a and b, and encrypt the message “20” and then decrypt it back to get the original message.

Prove that if p, q are positive integers such that p|q and q|p, then p = q