Logical Proofs - Valid Equivalence Arguments

Question 1

Double Negation (DN):

Answer 1

p :: ~ ~ p

Question 2

p :: ~ ~ p

Answer 2

Double Negation (DN):

Question 3

DeMorgan’s Theorem (DeM):

Answer 3

~ (p . q) :: ~ p ∨ ~ q

~ ( p ∨ q) :: ~ p . ~ q

Question 4

~ (p . q) :: ~ p ∨ ~ q

~ ( p ∨ q) :: ~ p . ~ q

Answer 4

DeMorgan’s Theorem (DeM):

Question 5

Commutation (Comm):

Answer 5

p ∨ q :: q ∨ p

p . q :: q . p

Question 6

p ∨ q :: q ∨ p

p . q :: q . p

Answer 6

Commutation (Comm):

Question 7

Association (Assoc):

Answer 7

p ∨ (q ∨ r) :: (p ∨ q) ∨ r

p . (q . r) :: (p . q) . r

Question 8

p ∨ (q ∨ r) :: (p ∨ q) ∨ r

p . (q . r) :: (p . q) . r

Answer 8

Association (Assoc):

Question 9

Distribution (Dist):

Answer 9

p . (q ∨ r) :: (p . q) ∨ (p . r)

p ∨ (q . r) :: (p ∨ q) . (p ∨ r)

Question 10

p . (q ∨ r) :: (p . q) ∨ (p . r)

p ∨ (q . r) :: (p ∨ q) . (p ∨ r)

Answer 10

Distribution (Dist):

Question 11

Contraposition (Contra):

Answer 11

p ⊃ q :: ~ q ⊃ ~ p

Question 12

p ⊃ q :: ~ q ⊃ ~ p

Answer 12

Contraposition (Contra):

Question 13

Implication (Impl):

Answer 13

p ⊃ q :: ~ p ∨ q

Question 14

p ⊃ q :: ~ p ∨ q

Answer 14

Implication (Impl):

Question 15

Exportation (Exp):

Answer 15

(p . q) ⊃ r :: p ⊃ (q ⊃ r)

Question 16

(p . q) ⊃ r :: p ⊃ (q ⊃ r)

Answer 16

Exportation (Exp):

Question 17

Tautology (Taut):

Answer 17

p :: p . p
p :: p ∨ p

Question 18

p :: p . p
p :: p ∨ p

Answer 18

Tautology (Taut):

Question 19

Equivalence (Equiv):

Answer 19

p ≡ q :: (p ⊃ q) . (q ⊃ p)
p ≡ q :: (p . q) ∨ (~ p . ~ q)

Question 20

p ≡ q :: (p ⊃ q) . (q ⊃ p)
p ≡ q :: (p . q) ∨ (~ p . ~ q)

Answer 20

Equivalence (Equiv):

Question 21

Rules of Equivalence:

Answer 21

Bi-directional

Question 22

Rules of Equivalence ______ be used with a single part of a statement

Answer 22

CAN