1. Propositional Equivalences
Section 1.2

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

3. 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

4. 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

5. 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.

6. Important Equivalences
Identity
p  T  p
p  F  p
Domination
p  T  T
p  F  F
Idempotent
p  p  p
p  p  p
Double Negation
( p)  p

7. Important Equivalences
Commutative
p  q  q  p
p  q  q  p
Associative
(p  q)  r  p  (q  r)
(p  q)  r  p  (q  r)
Distributive
p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p  r)
De Morgan’s
(p  q)  p  q
(p  q)  p  q

8. Important Equivalences
Absorption
p  (p  q)  p
p  (p  q)  p
Negation
p  p  T
p  p  F

9. Example
Show that (p  (p  q)) and p  q are logically equivalent.

10. Important Equivalences Involving Implications
p → q  p  q
p → q   q →  p
(p → q)  (p → r)  p → (q  r)
(p → q)  (p → r)  p → (q  r)
p↔ q  (p → q)  (q → p)

11. Example
Show that (p  q)  (p  q) is a tautology.

12. Next Lecture
1.3 Predicates and Quantifiers