Why Logic? A proof of any form requires logical reasoning. Logical reasoning ensures that the conclusions you reach are TRUE - as long as the rest of.

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Presentation transcript:

Why Logic? A proof of any form requires logical reasoning. Logical reasoning ensures that the conclusions you reach are TRUE - as long as the rest of the statements in the argument are also TRUE.

 For example:  All Mustangs are Fords. This fact can be represented by Venn diagram.

From the Venn diagram, we can also write an ”if-then” statement. If… Then… These If-Then statements are called conditional statements.

 In logical notation, conditionals are written as follows:  If p then q Or p q ( read as “p implies q”)

 In conditional, the part following the word if is the hypothesis. The part following the then word is the conclusion.  Identify the hypothesis and conclusion: If a car is a Mustang, then it is a Ford.

 Write the statement as a conditional. Underline the hypothesis and circle the conclusion. Also draw a Venn diagram for the statement. North Thurston HS is in Washington.

 Now consider the following statement: You attend NTHS. By placing YOU into our Venn diagram, what can you logically conclude?

 When you switch the hypothesis and conclusion of a conditional statement, you have the CONVERSE of the conditional. Example: Write the converse of the conditional Conditional: If you have a dog, then you have a pet. Converse:

 When you negate the hypothesis and conclusion of the conditional statement, you have the INVERSE of the conditional. Example: Write the inverse of the conditional Conditional: If you have a dog, then you have a pet. Inverse:

 When you switch AND negate the hypothesis and conclusions statement, you have the CONTRAPOSITIVE of the conditional. Example: Write the contrapositive of the conditional Conditional: If you have a dog, then you have a pet.

 In the previous example, the conditional statement is true. Are the related conditionals true?  Converse?  Inverse?  Contrapositive?  How did you know?

 The contrapositive of a true statement is always TRUE, and the contrapositive of a false condition is always FALSE.  The converse and inverse of a conditional are either both TRUE or both FALSE.  An example which proves that a statement is false is a COUNTEREXAMPLE.

Write the converse, inverse, and contrapositive for the conditional. Determine if the statements are true or false. If false, give a counterexample. If you are 16 years old, then you are a teenager.

 Conditional statements that can be linked together are called LOGICAL CHAINS. An example of a logical chain is the children’s series “If you give..”

 Arrange the following conditionals into a logical chain. Given: 1) If there is a parade, then fireworks will go off. 2) If there is July 4 th, then flags are flying. 3) If flags are flying, then there is a parade. Prove: If there is July 4 th, then fireworks will go off.