Chapter 1: The Foundations:
Logic and Proofs
1.1 Propositional Logic
1.2 Propositional Equivalences
1.3 Predicates and Quantifiers
1.4 Nested Quantifiers
1.5 Rules of Inference
1.6 Introduction to Proofs
1.7 Proof Methods and Strategy
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1.2: Propositional Equivalences
Definition:
Tautology: A compound proposition that
is always true.
Contradiction: A compound proposition
that is always false.
Contingency: A compound proposition
that is neither a tautology nor a
contradiction.
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Logical Equivalences
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Compound propositions that have
the same truth values in all possible
cases are called logically equivalent.
• Definition:
The compound propositions p and q are
called logically equivalent if pq is a
tautology. Denote pq.
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Logical Equivalences
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•
One way to determine whether
two compound propositions are
equivalent is to use a truth table.
Symbol: PQ
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Logical Equivalences
•
Prove the De Morgan’s Laws.
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Logical Equivalences
• HW: Prove the other one (De Morgan’s
Laws).
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Logical Equivalences
• Example:
Show that pq and ¬pq are logically
equivalent.
• HW: example 4 of page 23
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Logical Equivalences
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Logical Equivalences
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Logical Equivalences
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Logical Equivalences
Example 5: Use De Morgan’s laws to express the
negations of “Miguel has a cellphone and he
has a laptop computer”.
Example 5: Use De Morgan’s laws to express the
negations of “Heather will go to the concert or
Steve will go to the concert”.
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Logical Equivalences
• Example 6: Show that ¬(pq) and p ¬q are
logically equivalent.
• Example 7: Show that ¬(p(¬p  q)) and ¬p 
¬q are logically equivalent by developing a
series of logical equivalences.
• Example 8: Show that (p  q) ( pq) is a
tautology.
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Terms
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Tautology
Contradiction
Contingency
Logical Equivalence
De Morgan’s Laws
Commutative Law
Associative Law
Distributive Law
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