Answer to Question #314259 in Discrete Mathematics for pauljohn

Question

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 )

Answer 1

1

a) It is below freezing and it is snowing.

b) It is below freezing and it is not snowing.

c) It is not below freezing and it is not snowing.

d) It is snowing or it is below freezing.

e) If it is below freezing then it is snowing.

f) It is snowing and it is not below freezing.

g) If it is snowing then it is below freezing.

2

a)

b)

c)

d)

Hence a contradiction. Because the truth values are all False

e)

The proposition is logically equivalent as they have the same truth values.