Laws of Equivalence

Question 1

P ↔ P

Answer 1

Reflexitivity

Question 2

~(~P) ↔ P

Answer 2

Double Negation

Question 3

(P ∧ Q) ↔ (Q ∧ P)
(P ∨ Q) ↔ (Q ∨ P)
(P ↔ Q) ↔ (Q ↔ P)

Answer 3

Commutativity

(The position of the propositions can be interchanged, given that they are connected via conjunction (‘AND’ ^), disjunction (‘OR’ V), or biconditional (‘IF AND ONLY IF’)

Question 4

[(P ∧ Q) ∧ R] ↔ [P ∧ (Q ∧ R)]
[(P ∨ Q) ∨ R] ↔ [P ∨ (Q ∨ R)]
[(P ↔ Q) ↔ R] ↔ [P ↔ (Q ↔ R)]

Answer 4

Associativity

(The groupings of the propositions can be interchanged, given that they are all connected with the same logical connectives, which can be a conjunction (‘AND’ ^), disjunction (‘OR’ V), or biconditional (‘IF AND ONLY IF’)

Question 5

(P ∧ P) ↔ P
(P ∨ P) ↔ P

Answer 5

Idempotency

(A proposition conjoined or disjoined to itself, is just the preposition itself)

Question 6

(P ∧ T) ↔ P
(P ∨ F) ↔ P

Answer 6

Identity

(A proposition conjoined with to True statement is just the proposition itself since it will be the one to dictate the overall truth value of the compound preposition.)

(The same reasoning applies when a proposition is disjoined to False)

Question 7

(P ∧ ~P) ↔ F
(P ∨ ~P) ↔ T

Answer 7

Inverse

(A proposition conjoined with the negation of itself will always be FALSE)

(While a proposition disjoined to its negation will always be TRUE)

Question 8

(P ∧ F) ↔ F
(P ∨ T) ↔ T

Answer 8

Dominance

(A proposition does not hold any power in changing the Truth Value of the compound proposition)

Question 9

[P ∧ (P ∨ Q)] ↔ P
[P ∨ (P ∧ Q)] ↔ P

Answer 9

Absorption

A proposition (P) conjoined to a disjunction of itself to another proposition is just the proposition itself

The same can be said whenever P is disjoined to a conjunction of itself to another proposition.

Question 10

~(P ∧ Q) ↔ (~P ∨ ~Q)
~(P ∨ Q) ↔ (~P ∧ ~Q)

Answer 10

De Morgan’s Law

Negation can be distributed to the propositions inside the parenthesis, but the conjunction will be changed to a disjunction and vice versa.

Question 11

(P →Q) ↔ (~Q → ~P)

Answer 11

Contrapositive

Question 12

(P →Q) ↔ (~P ∨ Q)

Answer 12

Material Implication

P implies Q is just equal to the negation of the premise (P) disjoined to its conclusion (Q).

Question 13

(P ↔Q) ↔ [(P → Q) ∧ (Q → P)]
(P ↔Q) ↔ [(P ∧ Q) ∨ (~P ∧ ~Q)]

Answer 13

Material Equivalence

P if and only if Q is just equal to P implies Q or Q implies P

It is also just equal to P and Q or not P and not Q.

Question 14

[(P ∧ Q) → R] ↔ [P → (Q → R)]

Answer 14

Exportation

(P and Q) implies R is just equal to P implies (Q implies R).