Math 12 Foundations
3.5 Conditional Statements and Their Converse
By the end of the lesson you will be able to:
1. Understand and interpret:
a. conditional statements (⇒)
b. converse statements
c. biconditional statements (⟺)
2. Create and verify conditional, converse and biconditional statements.
Consider these statements:
1. If Brittany is texting, then she is using a cellphone.
2. If Brittany is using a cellphone, then she is texting.
These are both _________________________________, sometimes referred to as “If-Then” statements.
Conditional statements are composed of two parts:
a) ______________________________ (p) - an assumption
b) ______________________________ (q) - the result of the hypothesis
We can read this as p implies q or p ⇒ q.
Statement #2 is the _____________________________ of statement #1, in which the
____________________ and the ___________________________ are switched.
Example 1: Verifying a conditional statements
Eric’s soccer coach makes a conditional statement about soccer practice:
“If it is raining outside and we practice indoors.”
We can verify by checking all possible cases:
Case 1: The hypothesis and the conclusion are true.
It does rain outside and we practice indoors.
p
Case 2: The hypothesis is false and the conclusion is false.
It does not rain outside and we practice outdoors.
Case 3: The hypothesis is false and the conclusion is true.
It does not rain outside, we practice indoors.
Case 4: The hypothesis is true and the conclusion is false.
It does rain outside and we practice outdoors.
q
p⇒q
Math 12 Foundations
Another way to disprove a conditional statement is to provide a counterexample:
“If an animal has 4 legs, it must be a cow.”
Counterexample:
Example 2: Determining if a statement is biconditional
“A person who cannot distinguish between certain colours
is colour blind”
a) Is the conditional statement true?
A biconditional statement is a conditional
statement where the __________ is also true.
We can then say write this as:
“p ___________________ q” or “p ______ q”
For example: “If a number is even, then it is
divisible by 2.”
b) What is the converse statement? Is it true?
The converse of this statement is also true: “If
a number is divisible by 2, then it is even.”
Therefore, this statement is biconditional and
we write “A number is even if and only if it is
divisible by 2.”
c) Is the statement biconditional?
Assignment: P. 203 #1, 5, 6, 8ac, 14