Session 3: Logical Equivalences
• Showing Logical Equivalence
• Important Logical Equivalences
• Contrapositive, Converse and Inverse
• Equivalence Proofs
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Logical Equivalence
Two compound propositions p and q are logically equivalent if p ↔ q
is a tautology
We write this as p ≡ q where p and q are compound propositions
Two compound propositions p and q are equivalent if and only if the
columns in a truth table giving their truth values agree
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Example
Using a truth table we show that ¬p ∨ q ≡ p → q
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De Morgan’s Laws
Using a truth table show that De Morgan’s Second Law holds
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Equivalences with Basic Connectives
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Equivalences with Implications
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Contrapositive, Converse and Inverse
¬q → ¬ p is the contrapositive of p → q,
𝑝 → 𝑞 ≡ ¬𝑞 → ¬𝑝
From p → q we can form the following conditional statements
• q→p
• ¬p→¬q
is the converse of p → q
is the inverse of p → q (and the contrapositive of q → p)
They are equivalent to each other, but not equivalent to p → q
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Applying Logical Equivalences
The propositions in a known equivalence can be replaced by any
compound proposition
Example: since we know that
𝑝 → 𝑞 ≡ ¬𝑞 → ¬𝑝
we also know that, for example
(𝑝1 ∨ 𝑝2) → (𝑞1 ∧ 𝑞2) ≡ ¬ (𝑞1 ∧ 𝑞2) → ¬ (𝑝1 ∨ 𝑝2)
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Constructing New Logical Equivalences
• We can show that two expressions are logically equivalent by
developing a series of logically equivalent statements
• To prove that A ≡ B we produce a series of equivalences beginning
with A and ending with B
• This is called an equivalence proof
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Example: Equivalence Proofs
Show that
is logically equivalent to
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Example: Equivalence Proofs
Example: Show that
Advanced Information, Computation, Communication - 1. Logic and Proofs
is a tautology.
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Summary
• Showing Logical Equivalence
• De Morgan’s Laws
• Many Logical Equivalences
• Contrapositive, Converse and Inverse
• Equivalence Proofs
• Next: Normal Forms
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