Logical Equivalences with Conditionals

Question 1

p -> q

2 answers

Answer 1

= ¬p v q

= ¬q -> ¬p

Question 2

p v q

Answer 2

= ¬p -> q

Question 3

p ^ q

Answer 3

= ¬(p -> ¬q)

Question 4

¬(p -> q)

Answer 4

= p ^ ¬q

Question 5

(p -> q) ^ (p -> r)

Answer 5

= p -> (q ^ r)

Question 6

(p -> r) ^ (q -> r)

Answer 6

= (p v q) -> r

Question 7

(p -> q) v (p -> r)

Answer 7

= p -> (q v r)

Question 8

(p -> r) v (q -> r)

Answer 8

= (p ^ q) -> r

Question 9

p <=> q

3 answers

Answer 9

= (p -> q) ^ (q -> p)
= ¬p <=> ¬q
= (p ^ q) v (¬p ^ ¬q)

Question 10

¬(p <=> q)

Answer 10

= p <=> ¬q