Foundations of Math 12
KIM
3.5 Conditional Statements
Today’s Goal: To Understand and Interpret conditional statement
Statements: A statement is a sentence that is either true or false. A sentence that may be judged
true by one person and false by another is NOT considered a statement- it is an opinion.
Eg. Vancouver is the most beautiful city in North America.
Not a statement – Opinion
Eg. Vancouver is the most expensive city in North America, according to rankings by The
Economist.
Conditional Statements: a statement that is written using “if” and then”
-has a hypothesis/ assumption (follows “if”)
-has a conclusion/ result (follows “then”)
-may be true or false
Eg. If it is raining outside, then we practice indoors.
Verifying the conditional statement:
p
q
p⇒q
hypothesis
Conclusion
True Statement?
T
T
T
It rains outside, and they practices indoors
T
F
F
It rains outside and they do not practice indoors
(only time the conditional statement is false)
F
T
T
It does not rain and they practice indoors
F
F
T
It does not rain and they do not practice indoors
Foundations of Math 12
KIM
Converse:
- formed by interchanging the ‘if and then’ statement in a conditional statement.
Eg. If the team practice indoors, then it is raining outside.
Is the converse true? FALSE
Biconditional Statement:
-both the conditional statement and its converse are true
Eg.
If a number is divisible by 2, then it is even.
Converse
If the number is even, then it is divisible by 2.
Therefore, a number is even if and only if it is divisible by 2.
Symbol: p ⇔ q iff
Inverse:
-formed by negating both the hypothesis and the conclusion of a conditional statement
Eg. If a number is not divisible by 2, then it is not even.
Contrapositive:
-formed by taking the converse of the conditional statement and negating both the
hypothesis and the conclusion.
Eg.
If the number is not even, then it is not divisible by 2.
If they do not practice indoors, then it does not rain outside.
-Contrapositive and Conditional statement are always same either both true /both false.
Foundations of Math 12
KIM
Ex. Consider the following statements. For each,
a) Write the conditional statement and determine whether it is true.
b) Write the converse and determine whether it is true.
c) State whether the statement is biconditional.
d) Write the inverse and determine whether it is true.
e) Write the contrapositive statement and determine whether it is true.
1. A number that is divisible by 5 has a 0 as its final digit.
a) If a number is divisible by 5, then the number has a 0 as its final digit.
False
ex. 25 -> counter example
b) If a number has a 0 as its final digit, then the number is divisible by 5. True
c) Only converse is true. no, it is not biconditional
d) If a number is not divisible by 5, then the number does not have a 0 as its final digit. T
e) If a number does not have a 0 as its final digit, then the number is not divisible by 5.
False
2. An equilateral triangle has three equal sides.
a) If a triangle is an equilateral, then it has three equal sides.
T
○
b) If a triangle has three equal sides, then it is equilateral.
T
○
c) biconditional
both true A triangle is equilateral if and only if it has 3 equal sides.
d) If a triangle is not an equilateral, then it does not have three equal sides.
d) If a triangle does not have three equal sides, then it is not equilateral. T
HW: Textbook Pg. 203-204 #1, 4, 3, 13 Pg. 215 #1, 3, 5, 6, 9, 10 REVIEW Next Class
CH. Test on Friday
T
Foundations of Math 12
In-class Problem
KIM
November 21
/12
Name:
1. Write each statement as a conditional statement, and determine whether each conditional statement is
true or false. Give a counter example for any false statement.
a) A quadrilateral that is a square has diagonals that bisect each other.
If a quadrilateral is a square, then the diagonals of the quadrilateral bisect each other.
T
b) Students who attend high school in North Vancouver attend high school in British
Columbia.
If Students attend high school in N.V., then they attend high school in B.C.
T
c) A polygon with 6 sides is a hexagon.
If a polygon has 6 sides, then the polygon is a hexagon.
T
2. Write the inverse of each of the conditional statements in question #1, and determine whether each
inverse is true or false.
a) If a quadrilateral is NOT a square, then the diagonal of the quadrilateral DOES NOT
bisect each other. F
b) If students do not attend high school in N.V. then they do not attend high school in B.C. F
c) If a polygon does not have 6 sides, then the polygon is not a hexagon T
3. Write the converse of each of the conditional statements in question #1, and determine whether each
converse is true or false. Provide a counter example for each false statement.
a) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a square.
False
Rectangle
b)If the students attend high school in B.C., then the students attend high school in N.V.
F Students attend high school in Burnaby
c)If a polygon is a hexagon, then it has 6 sides.
T
4. Write the contrapositive of each of the conditional statements in question #1, and determine whether
each contrapositive statement is true or false.
a)If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a
square.
T
b)If students do not attend high school in B.C., then the students do not attend high school
in N.V.
T
c) If a polygon is not a hexagon, then it does not have 6 sides. T