Logical Form and
Logical Equivalence
Lecture 2
Section 1.1
Fri, Jan 19, 2007
Statements


A statement is a sentence that is either true or
false, but not both.
These are statements:



It is Wednesday.
Discrete Math meets today.
These are not statements:



Hello.
Are you there?
Go away!
Logical Operators

Binary operators



Unary operator


Conjunction – “and”.
Disjunction – “or”.
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 Wednesday.”
q = “Discrete Math meets today.”
Compound statements



p  q = “It is Wednesday and Discrete Math meets
today.”
p  q = “ It is Wednesday or Discrete Math meets
today.”
p = “It is not Wednesday .”
False Negations

Statement


False negation


Everyone likes me.
Everyone does not like me.
True negation

Someone does not like me.
False Negations

Statement


False negation


Someone likes me.
Someone does not like me.
True negation

No 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 is true if p is true and q is true.
 p  q is false if p is false or q is false.

p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F
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q
T
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
DeMorgan’s Laws:
(p  q)  (p)  (q)
(p  q)  (p)  (q)
 If it is not true that

i < size && value != array[i]
then it is true that…
DeMorgan’s Laws
DeMorgan’s Laws:
(p  q)  (p)  (q)
(p  q)  (p)  (q)
 If it is not true that

i < size && value != array[i]
then it is true that
i >= size || value == array[i]
DeMorgan’s Laws

If it is not true that
x  5 or x  10,
then it is true that …
DeMorgan’s Laws

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.


It is a logical form that is true for all logical
values of its variables.
A contradiction is a statement that is
logically equivalent to F.

It is a logical form that is false for all logical
values of its variables.
Tautologies and
Contradictions

Some tautologies:
p  p
 p  q  (p  q)


Some contradictions:
p  p
 p  q  (p  q)
