Discrete Mathematics Answers

Determine whether each of these pairs of sets are equal.

a. {1,3,3,3,5,5,5,5},{5,3,1}

b. {{1}}{1,{1}}

c. ∅,{∅}

Q.2- 1) For the given relation R = {(1,1), (1,2), (2,3), (3,1), (3,2)} defined on set A = {1,2,3}, find its transitive closure, R* using Warshall’s algorithm *

Show, by the use of the truth table (truth matrix), that the (p v q) v [(¬p) ʌ (¬q)]  is a contradiction. 

Show that ¬p →(q → r) and q → (p V r) are logically equivalent.

(a) Use Euclidean algorithm to find the gcd of 105 and 231.

(b) Use mathematical induction to show that

1 · 1! + 2 · 2! + · · · + n · n! = (n + 1)! − 1

(c) Show that the relation ∼ defined on R as a ∼ b if b−a ∈ Q, is an equivalence relation.

Also, find the equivalence class of 1.

Write the negation in English.

a. There is a student s such that for all courses x, s like x

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Everyone in your class has a cellular phone. b) Somebody in your class has seen a foreign movie. c) There is a person in your class who cannot swim. d) All students in your class can solve quadratic equations. e) Some student in your class does not want to be rich

Let P(x) denote the statement “x ≤ 4.” What are these truth values?

a) P(0)

b) P(4)

c) P(6)

d) P(5)

e) P (1)

Suppose that A = {1,3,5}, B = {1,5}, C = {3,7}, and D = {1,3}. Determine which of these sets are subsets of

which other of these sets.

How many 10-bit strings contain 6 or more 1’s? Explain and show all the steps.