Propositional Equivalences

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Propositional Equivalences

• Section 1.2

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Example

• You cannot ride the roller coaster if you are
  under 4 feet tall unless you are older than 16
  years old.

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Basic Terminology

• A tautology is a proposition which is always
  true.
• p ? ?p
• A contradiction is a proposition that is always
  false.
• p ? ?p
• A contingency is a proposition that is neither a
  tautology nor a contradiction.
• p ? q ? ?r

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Logical Equivalences

• Two propositions p and q are logically equivalent
  if they have the same truth values in all
  possible cases.
• Two propositions p and q are logically equivalent
  if p ? q is a tautology.
• Notation p ? q or p ? q

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Determining Logical Equivalence

• Use a truth table.
• Show that ?(p ? q) and ?p ? ?q are logically
  equivalent.
• Not a very efficient method, WHY?
• Solution Develop a series of equivalences.

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Important Equivalences
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Important Equivalences
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Important Equivalences
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Example

• Show that ?(p ? (?p ? q)) and ?p ? ?q are
  logically equivalent.

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Important Equivalences Involving Implications
p ? q ? ?p ? q p ? q ? ? q ? ? p
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Example

• Show that (p ? q) ? (p ? q) is a tautology.

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Next Lecture

• 1.3 Predicates and Quantifiers