An important type of step used in a mathematical argument is the replacement of a statement with another with the same truth value. Because of this, methods that propositions with the same truth value as a given compound proposition are used extensively in the construction of mathematical arguments.

1. WENNILOUPORAZO
DISCRETE MATHEMATICS LECTURE 2
PROPOSITIONAL
EQUIVALENCES
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2. Tautology
Definition
A compound proposition that is always true, no
matter what the truth values of the propositional
variables that occur in it, is called a tautology.
discrete mathematics

3. contradiction
Definition
A compound proposition that is always false.
discrete mathematics

4. contingency
Definition
A compound proposition that is neither a
tautology nor a contradiction.
discrete mathematics

5. complete the table:
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p ∨ ¬p

6. problem 1:
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p ∨ ¬p

7. solution:
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p ∨ ¬p
T
T
T
T
Therefore, p ∨ ¬p is
a Tautology.

8. problem 2:
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p ∧ ¬p

9. solution:
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p ∧ ¬p
F
F
F
F
Therefore, p ∧ ¬p is
a Contradiction.

10. logical
equivalences
discrete mathematics
• Compound propositions that have the same truth values in all
possible cases are logically equivalent.
• The compound prepositions p and q are called logically
equivalent if p ↔ q is a tautology.
• The notation p ≡ q denotes that p and q are logically
equivalent.

11. NOTE:
The symbol ≡ is not a logical connective,
and p ≡ q is not a compound proposition but
rather is the statement that p ↔ q is a
tautology.
discrete mathematics
The symbol ⇔ is sometimes used instead ≡
of to denote logical equivalence.

12. ALSO:
The compound prepositions p and q
are equivalent if and only if the
columns giving their truth values
agree.
discrete mathematics

13. ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
DE MORGAN'S LAWS
DISCRETE MATHEMATICS
¬(P ∨ Q) ≡ ¬P ∧ ¬Q

14. problem 1:
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Show that ¬(p ∧ q) and ¬(p ∨ q) are
logically equivalent.

15. solution:
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Using a truth table

16. explanation:
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Because the truth values of the compound
propositions ¬(p ∨ q) and ¬(p ∧ q) agree for all
possible combinations of the truth values of p and
q, it follows that ¬(p ∨ q) ↔ ¬(p ∧ q) is a tautology
and that these compound propositions are logically
equivalent.

17. Proof that ¬(p ∨ q) ↔ ¬(p ∧ q) is a tautology
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18. LOGICAL EQUIVALENCES
DISCRETE MATHEMATICS

19. LOGICAL EQUIVALENCES
DISCRETE MATHEMATICS

20. LOGICAL EQUIVALENCES
DISCRETE MATHEMATICS

21. Show that ¬(p → q) and p ∧ ¬q are logically equivalent.example1:
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Solution:
¬(p → q) ≡ ¬(¬p ∨ q) by previous example
≡ ¬(¬p) ∧ ¬ q De Morgan's Law
≡ p ∧ ¬ q Double Negation Law