Logical Form and Logical Equivalence

Lecture 1

Section 1.1

Wed, Jan 12, 2005

Statements

A statement is a sentence that is either true or false, but not both.

Statements:It is raining.I am carrying an umbrella.

Not statementsHello.Are you there?Go away!

Logical Operators

Binary operators Conjunction – “and”. Disjunction – “or”.

Unary operator Negation – “not”.

Other operators XOR – “exclusive or” NAND – “not both” NOR – “neither”

Logical Symbols

Statements are represented by letters: p, q, r, etc.

means “and”. means “or”. means “not”.

Examples

Basic statements p = “It is raining.” q = “I am carrying an umbrella.”

Compound statements p q = “It is raining and I am carrying an umbrella.” p q = “ It is raining or I am carrying an umbrella.” p = “It is not raining.”

False Negations

StatementEveryone likes me.

False negationEveryone does not like me.

True negationSomeone does not like me.

False Negations

StatementSomeone likes me.

False negationSomeone does not like me.

True negationNo one likes me.

Truth Table of an Expression

Make a column for every variable. List every possible combination of truth

values of the variables. Make one more column for the expression. Write the truth value of the expression for

each combination of truth values of the variables.

Truth Table for “and”

p q p q

T T T

T F F

F T F

F F F

p q is true if p is true and q is true. p q is false if p is false or q is false.

Truth Table for “or”

p q is true if p is true or q is true. p q is false if p is false and q is false.

p q p qT T T

T F T

F T T

F F F

Truth Table for “not”

p is true if p is false. p is false if p is true.

p p

T F

F T

Example: Truth Table

Truth table for the statement (p) (q r).

p q r (p) (q r )

T T T T

T T F F

T F T F

T F F F

F T T T

F T F T

F F T T

F F F T

Logical Equivalence

Two statements are logically equivalent if they have the same truth values for all combinations of truth values of their variables.

Example: Logical Equivalence

(p q) (p q) (p q) (p q)

p q (p q) (p q) (p q) (p q)

T T T T

T F F F

F T F F

F F T T

DeMorgan’s Laws

(p q) (p) (q) (p q) (p) (q) (C++) If it is not true that

i < size && value != array[i]

then it is true thati >= size || value == array[i]

If it is not true that x 5 or x 10, then it is true that x > 5 and x < 10.

Tautologies and Contradictions

A tautology is a statement that is logically equivalent to T.

A contradiction is a statement that is logically equivalent to F.

Some tautologies: p p p q (p q)

Some contradictions: p p p q (p q)