Conditional Statements
section 2-2
If a statement is not written in the form "if p, then q", you can rewrite it into
that form in order to identify the hypothesis and the conclusion more easily.
To do so, you must decide which part of the statement depends on the other.
problem # 1 - Identify the hypothesis and conclusion of each conditional statement.
a. If today is Thanksgiving Day, then today is Thursday.
b. If Marian lives in Los Angeles, then Marian lives in California.
c. A number is divisible by 3 if it is divisible by 6.
There are four possible ways to write a conditional statement:
Form
Example
If p, then q.
If an animal is a tiger, then it is a cat.
q if p.
It is a cat if an animal is a tiger.
p implies q.
An animal is a tiger implies it is a cat.
p only if q.
An animal is a tiger only if it is a cat.
When using a Venn diagram to show a relationship, the inner oval represents
the hypothesis, while the outer oval represents the conclusion.
parrots
birds
If an animal is a parrot, then it is a bird.
problem # 2 - Write an "if, then" conditional statement from each of the following:
a. An isosceles triangle has two congruent sides.
b. Two angles that are complementary are acute.
c. A rainy day implies that we carry an umbrella.
d. It is a good day if the sun shines.
e. We go on picnics only if it is the weekend.
We pass the test.
f.
We study hard.
Conditional statements have a truth value of either true (T) or false (F).
The truth value is false only when the hypothesis is true and the
conclusion is false. All other times the truth value is true.
Consider the conditional statement: If I get paid, then I'll take you to a movie.
case # 1 - I get paid and I take you to a movie. The truth value is T.
case # 2 - I get paid and I don't take you to a movie. The truth value is F.
case # 3 - I don't get paid and I take you to a movie. The truth value is T.
case # 4 - I don't get paid and I don't take you to a movie. The truth value is T.
To show that a conditional statement is false (F), you only need to find one
example where the hypothesis is true and the conclusion is false.
problem # 3 - Determine whether each conditional is true. If false, give a
counterexample.
a. If today is Sunday, then tomorrow is Monday.
b. If an angle is obtuse, then it has a measure of 100 .
c. If an odd number is divisible by 2, then 10 is a perfect square.
The negation of a statement p is "not p", written as
p.
The negation of a true statement is false and the negation of a false statement is true.
problem # 4 - Write the negation of each statement:
a. The number I am thinking of is a prime number.
b. Beatrice did not pass the test.
problem # 5 - Write the converse, inverse, and contrapositive of the conditional
statement “If an animal is a cat, then it has four paws.” Find the truth
value of each.
Related conditional statements that have the same truth value are called
logically equivalent statements. A conditional and its contrapositive are
logically equivalent, and so are the converse and inverse.
Consider the conditional statement “If Maria’s birthday is February 29, then she
was born in a leap year.”
The conditional is true (T) and the contrapositive is true (T).
The converse is true (F) and the inverse is true (F).
The logical equivalence of a conditional and its contrapositive is known as the
Law of Contrapositive.