Conditional Statements.  Conditional Statement: A statement that can be written in the form “If p then q.”  Every Conditional Statement has 2 parts:

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Presentation transcript:

Conditional Statements

 Conditional Statement: A statement that can be written in the form “If p then q.”  Every Conditional Statement has 2 parts:  Hypothesis: The part of the statement that follows the “if”  Conclusion: The part of the statement that follows the “then”  Converse: The statement formed by exchanging the hypothesis and the conclusion. Symbols p → q (Conditional) q → p (converse) Example: If you attend school on Friday, then you will take a test. Example: If you took a test on Friday, then you attended school.

DefinitionSymbols The inverse is the statement formed by negating the hypothesis and conclusion. ~p  ~q Example: Original Statement: “If you attend school on Friday, then you will take a test.” Inverse Statement: “If you do not attend school on Friday, then you will not take a test.”

DefinitionSymbols The contrapositive is the statement formed by both exchanging and negating the hypothesis and conclusion. ~q  ~p Example: Original Statement: “If you attend school on Friday, then you will take a test.” Contrapositive Statement: “If you do not take a test on Friday, then you did not attend school.” “If you do not take a test on Friday, then you did not attend school.”

 *Remember, the hypothesis comes after the “if” and the conclusion comes after the “then”.  Identify the hypothesis and the conclusion of this conditional statement:  If the University of Texas won the 2006 Rose Bowl football game, then Texas was college football’s 2005 National Champion.  Hypothesis= “the University of Texas won the 2006 Rose Bowl football game”  Conclusion= “Texas was college football’s 2005 National Champion”

 Identify the hypothesis and conclusion of the following statement:  If, T – 42 = 6, then T=48.  Hypothesis:  Conclusion: T – 42 = 6 T = 48 Good Job!!

 Write each sentence as a conditional:  1. A rectangle has four right angles.  If a figure is a rectangle, then it has four right angles.  2. A tiger is an animal.  If something is a tiger, then it is an animal.  Hint:  Always read your conditional statement after you write it….  Is it true? Always?  If not, rewrite it  And…remember  The hypothesis comes after “if” and the conclusion comes after “then”.

 Write each sentence as a conditional…  1. An integer that ends with 0 is divisible by 5.  Which is the hypothesis? Which is the conclusion?  If an integer ends with 0, then it is divisible by 5  2. A square has four congruent sides.  If something is a square, then it has four congruent sides.

Example 5: Writing a Converse, Inverse and Contrapositive Steps: 1. Identify, or underline the hypothesis and conclusion. 2. Switch the hypothesis and conclusion. 1. Rewrite as the converse.  1. Write the converse of the following conditional:  If two lines intersect to form right angles, then they are perpendicular.  Hypothesis= “two lines intersect to form right angles”  Conclusion= “they are perpendicular”  If two lines are perpendicular, then they intersect to form right angles.

Example 5 (cont): Writing a Converse, Inverse and Contrapositive  Now, write the inverse of the following conditional:  If two lines intersect to form right angles, then they are perpendicular.  Hypothesis= “two lines intersect to form right angles”  Conclusion= “they are perpendicular”  If two lines are not perpendicular, then they do not intersect to form right angles.

Example 5 (cont): Writing a Converse, Inverse and Contrapositive  Now, write the contrapositive of the following conditional:  If two lines intersect to form right angles, then they are perpendicular.  Hypothesis= “two lines intersect to form right angles”  Conclusion= “they are perpendicular”  If two lines do not intersect to form right angles, then the lines are not perpendicular.

Section 2-2 (cont.)

Truth Value- True or False, show that every time the hypothesis is true, the conclusion is also true  To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. you need to find only one counterexample for which the hypothesis is true and the conclusion is false  To show that a conditional is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false.

Example 1: Find the truth value  If you are late, then you are on time. -FALSE, because when p is true, it does not give a true q. So if you are late, it is not possible to be on time.  If a human is a cat, then squares have corners.  True, even though p is false, q is still true. Squares will always have four corners! So regardless of the hypothesis, the conclusion is always true! In order to prove that this is false, we would have to find an example where the conclusion is false.

Example 2: Find the truth value  If an angle is obtuse, then it has a measure of 100 degrees.  False.  If an odd number is divisible by 2, then 8 is a perfect square.  False, the conclusion will NEVER be true.

Example 4: Finding a Counterexample Show that the conditional is false, by finding a counterexample.  1. If you live in California, then you live within 10 miles from the beach.  False, because I live in Yucaipa, California, which is 60 miles from the beach.  2. If you play football, then you like every sport.  False, because Joe plays football, but he does not like baseball.

Great!!  Prove the statement is false by writing a counterexample…  1. If the name of the state contains the word “New”, then the state borders an ocean.  False, New Mexico contains the word “New” and it does not border an ocean. You Try…

Use the Venn diagram below. What does it mean to be inside the large circle but outside the small circle? The large circle contains everyone who lives in California. The small circle contains everyone who lives in Anaheim. To be inside the large circle but outside the small circle means that you live in California but outside Anaheim. Conditional Statements Write a conditional statement for the Venn diagram above. If you are a resident of Anaheim, then you are a resident of California.

Assignment #18 Page 85 #3-5 a)-e) #9-12 #19-21 #34-36