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Boolean Laws

In Boolean Laws, Logical equivalence is a type of relationship between any two statements. If p and q are the two statements if they are logically equivalent mean p and q are same. The logical relation of any two statements is related by the statement “if and if only”. Let us consider the following statement,

If Rita is in America then she is in London

If Rita is not in America then she is not in London

Here Statement p`->` Rita is in America

q `->` She is in London

Where p and q is logically related.

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Laws of Logical Equivalence

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We will use the following laws for Logical Equivalence

  • De Morgan’s law
  • Distributive law and prepositions
  • Complement laws
  • Identity laws
  • Commutative laws
  • Associative laws

The logical Equivalence is indicated by ? OR <=>

De Morgan's Law: It is used to solve the logical equivalence.

¬ ( P V Q) ? (¬P)?(¬Q)

¬ ( P ? Q) ? (¬P)V(¬Q)

Distributive law: It is like normal arithmetic distributive law.

P?(Q V R) = (P ? Q) V (p ? R)

PV(Q ? R) = (P V Q) ? (P VR)

Complement Laws: Component law is used to perform the unary and binary operations in logical equivalence.

P V ¬P ? T

P ^ ¬P ?F

¬ ¬ P? T

¬ T ? F

¬ F ? T

More Laws of Logical Equivalence

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Identity laws:

P ? T ? P

P ? F ? F

P V T ? T

P V T ? P

Commutative laws:

P ? Q ? Q ? P

P V Q ? Q V P

Associative Laws:

P ? ( Q ? R ) ? (P ? Q) ? R

P V ( Q V R ) ? (P V Q) V R

Example for Logical Equivalence:

(P ? Q)<=> not ( P ? Q) V (P V Q) Law of implication

<=> (not P V not Q) V ( P V Q) De Morgan's law

<=>(not P V P) V (not Q V Q) Associative law

<=>T V T

<=>T

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