Download presentation

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

Similar presentations

© 2019 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google