An Introduction to Logic

1. An Introduction to Logic

2. 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.

3. Using Logical Reasoning • For example: • All Mustangs are Fords. This fact can be represented by Venn diagram.

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

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

6. More about Conditionals • 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.

7. Example • 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.

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

9. Reversing Conditionals • 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:

10. Inverses of Conditionals • 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:

11. Contrapositives of Conditionals • 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.

12. TRUTH OR LIES? • In the previous example, the conditional statement is true. Are the related conditionals true? • Converse? • Inverse? • Contrapositive? • How did you know?

13. Truth or Lies? • 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.

14. Your Turn! 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.

15. Logical Chains • 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..” http://www.graves.k12.ky.us/powerpoints/elementary/winaelliott.ppt

16. Example: • Arrange the following conditionals into a logical chain. Given: • If there is a parade, then fireworks will go off. • If there is July 4th , then flags are flying. • If flags are flying, then there is a parade. Prove: If there is July 4th, then fireworks will go off.