Transcript Logic ChAPTER 3 - University of Texas at Brownsville

LOGIC
CHAPTER 3
1
VARIATIONS OF THE
CONDITIONAL AND
IMPLICATIONS
3.4
2
VARIATIONS OF THE
CONDITIONAL
Variations of p →q
Converse:
q  p
Inverse:
~p  ~q
Contrapositive:
~q  ~p
p →q is logically equivalent to ~q → ~p
q →p is logically equivalent to ~p →~ q
3
EXAMPLES OF VARIATIONS
p: n is not an even number.
q: n is not divisible by 2.
Conditional: p→q
If n is not an even number, then n is not divisible by 2.
Converse: q→p
If n is not divisible by 2, then n is not an even number.
Inverse: ~p→~q
If n is an even number, then n is divisible by 2.
Contrapositive: ~q→~p
If n is divisible by 2, then n is an even number.
4
Equivalent
Equivalent
Conditional
Converse
Inverse
Contrapositive
p
q
p→q
q→p
~p → ~ q
~q → ~p
T
T
T
T
T
T
T
F
F
T
T
F
F
T
T
F
F
T
F
F
T
T
T
T
CONDITIONAL EQUIVALENTS
Statement
Equivalent forms
if p, then q
p→q
p is sufficient for q
q is necessary for p
p only if q
q if p
6
BICONDITIONAL EQUIVALENTS
Statement
Equivalent forms
p if and only if q p is necessary and
sufficient for q
p↔q
q is necessary and
sufficient for p
q if and only if p
7
EXAMPLES OF VARIATIONS
Given: h: honk
u: you love Ultimate
Write the following in symbolic form.
u  h
Honk if you love Ultimate.
u  h
If you love Ultimate, honk.
Honk only if you love Ultimate. h  u
A necessary condition for loving Ultimate is to honk. u  h
A sufficient condition for loving Ultimate is to honk h  u
To love Ultimate, it is sufficient and necessary that you honk.
or
u  h
h  u
8
TAUTOLOGIES AND CONTRADICTIONS
A statement that is always true is called a
tautology.
A statement that is always false is called a
contradiction.
9
EXAMPLE
Show by means of a truth table that the
statement p↔ ~ p is a contradiction.
p
~p
p↔ ~ p
T
F
F
T
F
F
10
IMPLICATIONS
The statement p is said to imply the
statement q, p  q, if and only if the
conditional p→q is a tautology.
11
EXAMPLE
Show that [( p  q )  p ]  q
p q p →q
(p →q) Λ p
q
[(p →q) Λ p] →q
T T
T F
T
F
T
F
T
F
T
T
F T
F F
T
T
F
F
T
F
T
T
END
12