Propositional Equivalences - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSE115/ENGR160 Discrete Mathematics 01/19/

1.3 Propositional equivalences

• Replace a statement with another statement with the same truth value
• For efficiency (speed-up) or implementation purpose (e.g., circuit design)

Logical equivalence

• p ≡ q (p q): the compound propositions p and q are logically equivalent if p ↔ q is a tautology
• Can use truth table to determine whether two propositions are equivalent or not

Example

5

• Show that ┐(p v q) and ┐p ˄ ┐ q are equivalent

Example

Example

Example

• Express the negation of “Heather will go to the concert or Steve will go to the concert”
• Negation:

Heather will not go to the concert AND Steve will not go to the concert.

De Morgan’s law: general form

• The first example above is known as the De Morgan’s law

Constructing new logical equivalences

• Show ┐ (p ˅ (┐ p ˄ q) ) ≡ ┐ p ˄ ┐ q

┐ (p ˅ (┐ p ˄ q) ) ≡ ┐ p ˄ (┐(┐ p ˄ q)) ≡ ┐ p ˄ (┐(┐ p) ˅ ┐q) ≡ ┐ p ˄ (p ˅ ┐q) ≡ (┐ p ˄ p ) ˅ (┐ p ˄ ┐q) ≡ F ˅ (┐ p ˄ ┐q) ≡ ┐ p ˄ ┐q

Limitations of proposition logic

• Proposition logic cannot adequately express the meaning of statements
• Suppose we know

“ Every computer connected to the university network is functioning property”

• No rules of propositional logic allow us to conclude “MATH3 is functioning property” where MATH3 is one of the computers connected to the university network (^) Docsity.com 17

1.4 Predicate and quantifiers

• Can be used to express the meaning of a wide range of statements
• Allow us to reason and explore relationship between objects
• Predicates : statements involving variables, e.g., “x > 3”, “x=y+3”, “x+y=z”, “computer x is under attack by an intruder”, “computer x is functioning property”

Example: x > 3

• The variable x is the subject of the statement
• Predicate “is greater than 3” refers to a property that the subject of the statement can have
• Can denote the statement by p(x) where p denotes the predicate “is greater than 3” and x is the variable
• p(x): also called the value of the propositional function p at x
• Once a value is assigned to the variable x, p(x) becomes a proposition and has a truth value