Transcript Chapter 1 The Logic of Compound Statements Section 1.1 Logical Form and Logical Equivalence.

Chapter 1
The Logic of Compound Statements
Section 1.1
Logical Form and Logical Equivalence
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 statements
– He has a driver’s license.
– Are you there?
–x+y>0
Logical Operators
• Binary operators
– Conjunction – “and”.
– Disjunction – “or”.
• Unary operator
– Negation – “not”.
• Other operators
– XOR – “one or the other but not both””
– 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.”
Examples
• But compound statements
– “it is not hot but it is sunny.”
– but in this case is ^ “and”
– p = “it is not hot”
– q = “it is sunny”
– expression: p ^ q
– “it is not hot and it is not sunny”
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
• (p  q)  (p)  (q)
• (p  q)  (p)  (q)
• It is not true that “John is short and he is
fat”, then it is true that “John is not short
or John is not fat”.
• 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)
Wrapup
• Quiz on Tuesday (Chapter 1)
• Homework due Thursday