Logical Equivalence

Two propositions are logically equivalent if they always have the same truth value in every scenario. In this page, we will study about logical equivalence with their laws, truth table and examples.

Definition of Logical Equivalence

Two propositions p and q are called logically equivalent if both statements have the same truth table or the statement p↔q is a tautology.

If p and q are logically equivalent, then it is denoted as p≡q.

Logical Equivalence Examples

Question: Verify that the following statements are logically equivalent.

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

Solution:

1.

p	q	p∨q	q∨p	(p∨q)↔(q∨p)
T	T	T	T	T
T	F	T	T	T
F	T	T	T	T
F	F	F	F	T

So we see that both p∨q and q∨p have the same truth table. Also, the proposition (p∨q)↔(q∨p) is a tautology. Therefore, we conclude that p∨q ≡ q∨p.

Logical Equivalence Laws

There are several propositional equivalence laws/property which are listed in the table below.

Property Name	Property
Commutative Property	p∨q ≡ q∨p p∧q ≡ q∧p
Associative Property	p∨(q∨r) ≡ (p∨q)∨r p∧(q∧r) ≡ (p∧q)∧r
Distributive Property	p∨(q∧r) ≡ (p∨q)∧(p∨r) p∧(q∨r) ≡ (p∧q)∨(p∧r)
Absorption Property	p∨(p∧q) ≡ p p∧(p∨q) ≡ p
Idempotent Property	p∨p ≡ p p∧p ≡ p
Negation Property	p∨~p ≡ T p∧~p ≡ F
De Morgan’s Laws	~(p∨q) ≡ ~p∧~q ~(p∧q) ≡ ~p∨~q
Involution property or Double negation	~(~p) ≡ p
Identity Property	p∧T ≡ p, p∧F ≡ F p∨T ≡ T, p∨F ≡ p

Related Articles: Mathematical Logic: Logical Connectives, Truth Table

Tautology in Logic

Solved Problems

Question: Without following truth table prove that the following statements are logically equivalent.

1. p∨(p∧q) ≡ p
2. ~(p∨q)∨(~p∧q) ≡ ~p
3. p↔q ≡ (p∨q)→(p∧q)
4. p→(q→r) ≡ p∧q → r
5. P ↔ q ≡ (p ∧ q) ∨ (~ p ∧ ~ q)
6. (~p∧(~q∧r))∨(q∧r)∨(p∧r) ≡ r

Answers:

1. p∨(p∧q)

≡ (p∧T)∨(p∧q), since p∧T≡p

≡ p∧(T∨q) [Distributive property]

≡ p∧T

≡ p

---

2. ~(p∨q)∨(~p∧q)

≡ (~p∧~q)∨(~p∧q) [De Morgan’s Law]

≡ ~p∧(~q∨q) [Distributive]

≡ ~p∧T

≡ ~p

---

3. p↔q

≡ (p→q)∧(q→p)

≡ (~p∨q)∧(~q∨p)

≡ ((~p∨q)∧~q)∨((~p∨q)∧p) [Distributive]

≡ (~p∧~q)∨(~q∧q)∨(~p∧p)∨(p∧q) [Distributive]

≡ (~p∧~q)∨F∨F∨(p∧q)

≡ (~p∧~q)∨(p∧q)

≡ ~(p∨q)∨(p∧q)

≡ (p∨q)→(p∧q)

---

4. p→(q→r)

≡ ~p∨(q→r) Apply the implication law A→B ≡ ~A∨B.

≡ ~p∨(~q∨r)

≡ (~p∨~q)∨r

≡ ~(p∧q)∨r by De Morgan’s Law

≡ (p∧q)→r

---

5. P ↔ q

≡ (p→q)∧(q→p)

≡ (~p∨q) ∧ (~q∨p)

≡ [~p∧(~q∨p)] ∨ [q∧(~q∨p)] Distributive Law

≡ [(~p∧~q)∨(~p∧p)] ∨ [(q∧~q)∨(q∧p)] Distributive Law

≡ [(~p∧~q)∨F] ∨ [F∨(q∧p)]

≡ (~p∧~q) ∨ (p∧q)

≡ (p∧q)∨(~p∧~q)