MMiG2 b – Converses and conditional statements

Essential Questions?

Converses Examples

A statement is a sentence that is either true or false, but not both. Some examples of statements are: Atlanta is the capital of Georgia. (true) The Atlanta Braves are a professional football team. (false) Some examples of non-statements are: Write your name on your paper. What did you eat for breakfast?

A conditional statement is a statement that can be rewritten in “if…then…” form. Some examples of conditional statements are: If today is Monday, then tomorrow is Tuesday. If I need to write a paper, then I will use a computer. If I am asleep, then I will dream. The hypothesis is the part of the statement that follows the word “if”. The conclusion is the part of the statements that follows the word “then”.

Example 1: Identify the hypothesis and the conclusion of the Example 1: Identify the hypothesis and the conclusion of the conditional statements. If a polygon has three sides, then it is a triangle. hypothesis: a polygon has three sides conclusion: it is a triangle If the sun is shining, then we will go swimming. hypothesis: the sun is shining conclusion: we will go swimming Note: The word “if” is not part of the hypothesis and the word “then” is not part of the conclusion.

Conditional statements are not always written in “if…then…” form, but we can rewrite them. Example: I breathe when I sleep. If-then form: If I am asleep, then I am breathing. Example: I buy what I like. If-then form: If I like something, then I will buy it. In the two examples above, the hypothesis is found after the words “when” and “what”. Notice that sometimes it is necessary to use different words for the statement to make sense.

Example 2:. Rewrite the following as an “if…then…” statement Example 2: Rewrite the following as an “if…then…” statement. “All acute angles measure less than ninety degrees.” Solution: If an angle is acute, then its measure is less than ninety degrees.

If the hypothesis and the conclusion are switched, then the new statement is known as the converse. Example: If it is cold outside, then I will wear a coat. hypothesis: it is cold outside conclusion: I will wear a coat Converse: If I wear a coat, then it is cold outside. hypothesis: I wear a coat conclusion: it is cold outside Notice that the hypothesis and conclusion were switched.

Example 3: Write the conditional statement in “if…then…” form Example 3: Write the conditional statement in “if…then…” form. Then write the converse of the statement. “Two angles are complementary if their measures are 15º and 75º.” Conditional: If the measures of two angles are 150 and 750 , then they are complementary. Converse: If two angles are complementary, then their measures are 150 and 750 .

A conditional statement and its converse do not always have the same truth value.

Biconditional If a Conditional Statement and its Converse both have a truth value of TRUE, then they can be written as a Biconditional Statement: Ex: If a polygon has three sides, then it is a triangle. (T) Converse: If a polygon is a triangle, then it has three sides. (T) Soooooo…

It can be written as a biconditional statement…. A polygon is a triangle, if and only if it has three sides!

Write the converse and then determine the truth value.

Answers: A B D B, not necessarily!

Inverse

Inverses Examples

The inverse of a conditional statement is formed by negating the hypothesis and the conclusion. Example: If an animal has four legs, then it is a dog. hypothesis: an animal has four legs conclusion: it is a dog Inverse: If an animal does not have four legs, then it is not a dog.

Example 1: Write the inverse of the following conditional statements. Conditional: If I have a smile on my face, then it is Friday. Inverse: If I do not have a smile on my face, then it is not Friday. Conditional: If I do not ride the bus, then I will not go to school. Inverse: If I ride the bus, then I will go to school. Notice if the conditional statement contains the word “not”, then it will not be in the inverse statement.

Example 2:. Write the inverse of the following conditional. statement Example 2: Write the inverse of the following conditional statement. Determine the truth value of each. Conditional: If two lines are perpendicular, then they will intersect. (this statement is true) Inverse: If two lines are not perpendicular, then they will not intersect. (this statement is false) Like the converse, the inverse does not necessarily have the same truth value as the conditional statement.

Answers B D A

Contrapositive – Contra “+”

Contrapositives Examples

The contrapositive of a conditional statement is formed by switching the hypothesis and conclusion AND negating each. Conditional: If the temperature is 0 degrees Celsius, then it is 32 degrees Fahrenheit. hypothesis: the temperature is 0 degrees Celsius conclusion: it is 32 degrees Fahrenheit Contrapositive: If it is not 32 degrees Fahrenheit, then the temperature is not 0 degrees Celsius.

Example 1: Write the contrapositive of the following statement. If a number is prime, then it is divisible only by one and itself. hypothesis: a number is prime conclusion: it is divisible only by one and itself Contrapositive: If a number is not divisible only by one and itself, then it is not prime.

Example 2: Write the contrapositive of the following statement. Exponents are added if monomials are multiplied together. First, rewrite the statement in “if…then…” form. If monomials are multiplied together, then the exponents are added. Then, switch and negate the hypothesis and conclusion. If the exponents are not added, then the monomials are not multiplied together.

A statement’s inverse and converse may or may not have the same truth value as the original statement, but the contrapositive will always have the same truth value as the conditional statement. Example: Write the contrapositive of the following statement. Determine the truth value of each. Conditional: If the measure of an angle is 90 degrees, then it is a right angle. (true) Contrapositive: If an angle is not a right angle, then its measure is not 90 degrees. (true)

Example 3:. Write the converse, inverse, and contrapositive of Example 3: Write the converse, inverse, and contrapositive of the following statement. Determine the truth value of each. If a polynomial has three terms, then it is a trinomial. (true) Converse: If a polynomial is a trinomial, then it has three terms. (true) Inverse: If a polynomial does not have three terms, then it is not a trinomial. (true) Contrapositive: If a polynomial is not a trinomial, then it does not have three terms. (true)

Example 4:. Write the converse, inverse, and contrapositive of Example 4: Write the converse, inverse, and contrapositive of the following statement. Determine the truth value of each. Converse: Inverse: Contrapositive: Notice that the conditional statement and the contrapositive are logically equivalent (same truth value). Also, the converse and the inverse are logically equivalent.

a. If a number is not divisible by 2, then it is divisible by 4.

Answers: 1. C 2. D 3. D 4. D

More Practice

More Practice If I get done with my homework, then I will go to the game. If the temperature is 10°F, then the water will freeze. If a student has at least a 90 average, then the student is on the high honor roll.

Visit to Washington D.C., but NOT visit the Jefferson Monument. You could visit Springfield, Missouri!! If x = 1, then x2 is NOT greater than x!

p  q If you like to be at the beach, then you like volleyball. ~p  ~q If you do not like volleyball, then you do not like the beach. ~q  ~p If you do not like the beach, then you do not like volleyball.

If an angle measures less then 90°, then it is acute. If an angle is acute, then it measures less than 90°. If three points lie on the same line, then they are collinear. If three points are collinear, then they lie on the same line.

Converse: If you live in Michigan, then you live in Detroit. (F) Because the converse if false, this statement cannot be biconditional. You could live in Michigan and not live in Detroit. You could live in Lansing. Converse: If two angles have the same measure, then they are congruent. (T) Biconditional: Two angles are congruent, if and only if they have the same measure. Converse: If the sum of two angles measures 90°, then they are complementary. (T) Biconditional: Two angles are complementary, if and only if the sum of their measure is 90°.