Discrete Mathematics Answers

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) No one is perfect.

b) Not everyone is perfect.

c) All your friends are perfect.

d) At least one of your friends is perfect.

e) Everyone is your friend and is perfect.

f) Not everybody is your friend or someone is not perfect. Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives.

a) No one is perfect.

b) Not everyone is perfect.

c) All your friends are perfect.

d) At least one of your friends is perfect.

e) Everyone is your friend and is perfect.

f) Not everybody is your friend or someone is not perfect.

The number of transitive closure exists in the relation R = {(0,1), (1,2), (2,2), (3,4), (5,3), (5,4)} where {1, 2, 3, 4, 5} ∈ A is__________.

Suppose a, b, c, d have proper positions 1, 2, 3, 4 respectively, i.e., the correct sequence (from position 1 to 4) is a, b, c, d. Write down all the deranged sequences. What is the combinatorial expression for their count?

[￢p ∧(p ∨ q)]→q

For each of the ff. sets, determine whether 2 is an element of that set.

a. {𝑥∈ℝ|𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 1}

b. {𝑥∈ℝ|𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟}

c. {2,{2}}

d. {{2},{{2}}}

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.