Conditional Statements
Conditional Statements
If-then statements are called
conditional statements.
The portion of the sentence following if
is called the hypothesis. The part
following then is called the
conclusion.
p
q (If p, then q)
p
q
If it is an apple, then it is a fruit.
Hypothesis – It is an apple.
Conclusion – It is a fruit.
Converse q
p
The converse statement is formed
by switching the hypothesis and
conclusion.
If it is an apple, then it is a fruit.
Converse: If it is a fruit, then it is an
apple.
The converse may be true or false.
negation – the denial of a
statement
Ex. “An angle is obtuse.”
Negation – “An angle is not
obtuse.”
Inverse ~p
~q
An inverse statement can be formed
by negating both the hypothesis and
conclusion.
If it is an apple, then it is a fruit.
Inverse: If it is not an apple, then it is
not a fruit.
The inverse may be true or false.
Contrapositive ~q
~p
A contrapositive is formed by negating the
hypothesis and conclusion of the
converse.
If it is an apple, then it is a fruit.
Contrapositive: If it is not a fruit, then it is
not an apple.
The contrapositive of a true conditional is
true and of a false conditional is false.
If you double the radius of a circle, then you
will know the length of the diameter.
Write the given statements in symbolic form.
• If you don’t know the length of the diameter, Contrapositive
then you haven’t doubled the radius.
• If you don’t double the radius of a circle,
Inverse
then you will not know the length of the diameter.
Invalid
• If you know the length of the diameter,
then you haven’t doubled the radius.
• If you know the length of the diameter,
Converse
then you have doubled the radius.
• Select the statement(s) above that are not valid.
• Identify the converse, inverse, and contrapositive.
A biconditional statement is defined to be true whenever both parts have the same truth value. The biconditional operator is denoted by a double-headed arrow
Biconditional p  q or pq
• A biconditional statement is defined to be
true whenever both parts have the same
truth value. The biconditional operator is
denoted by a double-headed arrow . The
biconditional pq represents "p if and only if
q," where p is a hypothesis and q is a
conclusion..
.
T
Biconditional p  q
• p: a polygon is a triangle
• q: a polygon has exactly three sides.
• A polygon is a triangle if and only if it has
exactly three sides.
Biconditional p  q
• When a statement and the converse are both
true, the statements can form a biconditional
statement.
• p : 2x + 7 = 19
• q:x=6
• 2x + 7 = 19  x = 7
• 2x + 7 = 19 iff x = 7
The following is a truth table for biconditional
pq or pq
P
Q
PQ
T
T
T
T
F
F
F
T
F
F
F
T
Law of Detachment
• Law of Detachment ( also known as Modus
Ponens (MP) ) says that if pq is true and p
is true, then q must be true.
• pq The team who wins tonight’s game
wins the district championship.
CHS won the game the game.
• Therefore: CHS won the district
championship
Law of Syllogism
• The Law of Syllogism
( also called the Law of Transitivity ) states:
if p  q and q r are both true, then p  r
is true.
p  q : If you like raisins then you like fruit.
q r : If you like fruit, then you like sugar.
p  r : If you like raisins, then you like sugar.
Law of Contrapositive
• Why does a chicken cross the road?????
• A chicken crosses the road, because he
wants to get to the other side.
• Now let’s use the law of contrapositive to
make a valid argument.
• Why does that stupid chicken just sit
there???
• If a chicken doesn’t want to get to the other
side, then he doesn’t cross the road!
Counterexamples
• If you drive to school, then you have
money.
• If you have money, then you can buy
whatever you want.
∴ If you drive to school, then you
can buy whatever you want.
(Valid/False)
～ I drive to school but I can’t buy
my student’s interest in math.
I drive to school, but I can’t buy
Trump Towers.