Discrete Mathematics Answers

Show, by the use of the truth table (truth matrix), that are ¬(P∨(Q∨(¬P→¬R))) and ¬P(Q→R) logically equivalent. (15 points)

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

1. Let p and q be the propositions.

p: It is below freezing.

q: It is snowing.

Express each of these propositions in complete English sentences.

a) p ∧ q 

b) p ∧ ¬q 

c) ¬p ∧ ¬q 

d) q ∨ p 

e) p → q 

f) q ∧ ¬p 

g) q → p 

2. Solve the following.

a) Construct a truth table.

¬p ∧ ( p ↔ ¬q )

b) Construct a truth table.

p → ( q ∧ r )

c) Construct a truth table.

( p → q ) ∨ ( ¬p ↔ r )

d) Find out if the following is a tautology, contradiction, or contingency

( p ∨ q ) ∧ ( ¬p ∧ ¬q )

e) Find out if the following propositions have logical equivalence.

( p ↔ q ) ≡ ( p → q ) ∧ ( q → p )

show that ~p --> (q --> r ) and q --> (p v r) are logically equivalent

List the members of this set: {x | x is the square of an integer and x < 50}

Value of 3^222mod11

Let P(x) denote the statement x > 3. What is the truth value of the quan-

tification ∃xP(x), where the domain consists of all real numbers?

Construct a truth table (truth matrix) for each of these compound proposition

1. p ∧ ~q
2. p → ~q
3. ( p ∨ q ) → ( p ∧ q)
4. ( p ∨ q ) → r
5. ( p ∨ q )∧ ~r

by using truth table (truth matrix), show that each statement is a tautology, contradictory or a contingent statement.

1. ~p ∧ q
2. ~(p ∨ q) → q
3. ~(~p ∧ q) ∨ q
4. ~p → (p → q)
5. ~p → (p ∨ q)