Math 3336
Section 1.3
Propositional Equivalences
Topics:
• Tautologies, Contradictions, and Contingencies
• Logical Equivalence
• Important Logical Equivalences
• Showing Logical Equivalence
• Propositional Satisfiability
Tautologies, Contradictions, and Contingencies
Definitions:
A tautology is a compound proposition that is always true, no matter what the truth value of the
propositional variables that occur in it.
A contradiction is a compound proposition that is always false.
A contingency is a compound proposition that is neither a tautology nor a contradiction.
Example: Show that 𝑝𝑝 ∨ ¬𝑝𝑝 is a tautology, 𝑝𝑝 ∧ ¬𝑝𝑝 is a contradiction, and ¬𝑝𝑝 is a contingency.
Definition:
• Two compound propositions 𝑝𝑝 and 𝑞𝑞 are logically equivalent if 𝑝𝑝 ↔ 𝑞𝑞 is a tautology.
• We write this as 𝑝𝑝 ⇔ 𝑞𝑞 or 𝑝𝑝 ≡ 𝑞𝑞 where 𝑝𝑝 and 𝑞𝑞 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.
Example: Show that ¬𝑝𝑝 ∨ 𝑞𝑞 ≡ 𝑝𝑝 → 𝑞𝑞.
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De Morgan’s Laws
AUGUST DE MORGAN (1806 - 1871)
¬(𝑝𝑝 ∧ 𝑞𝑞) ≡ ¬𝑝𝑝 ∨ ¬𝑞𝑞
¬(𝑝𝑝 ∨ 𝑞𝑞) ≡ ¬𝑝𝑝 ∧ ¬𝑞𝑞
Example: Show that the second De Morgan law works.
Key Logical Equivalences
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More Logical Equivalences
Example: Establish the distributive law of disjunction over conjunction,
i.e. show that 𝑝𝑝 ∨ (𝑞𝑞 ∧ 𝑟𝑟) and (𝑝𝑝 ∨ 𝑞𝑞) ∧ (𝑝𝑝 ∨ 𝑟𝑟) are equivalent.
Constructing New Logical Equivalences
We can show that two expressions are logically equivalent by writing a series of logically
equivalent statements.
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Example (Equivalence Proof): Show that ¬(𝑝𝑝 ∨ (¬𝑝𝑝 ∧ 𝑞𝑞))is logically equivalent to ¬𝑝𝑝 ∧ ¬𝑞𝑞.
Example (Equivalence Proof): Show that (𝑝𝑝 ∧ 𝑞𝑞) → (𝑝𝑝 ∨ 𝑞𝑞) is a tautology.
Propositional Satisfiability
Definition: A compound proposition is satisfiable if there is an assignment of truth values to its
variables that make it true. When no such assignments exist, the compound proposition is
unsatisfiable.
• A compound proposition is unsatisfiable if and only if its negation is a tautology.
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Example: Determine the satisfiability of the following compound proposition
(𝑝𝑝 ∨ ¬𝑞𝑞) ∧ (𝑞𝑞 ∨ ¬𝑟𝑟) ∧ (𝑟𝑟 ∨ ¬𝑝𝑝)
Hint: Assign T to 𝑝𝑝, 𝑞𝑞, and 𝑟𝑟.
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