R e l a t i o n s h i p s A m o n g St a t e m e n t s

1. RelationshipsAmongStatements MM1G2.b. Understandand use the relationships among astatement and its converse, inverse, and contrapositive. BY: BETH MATHIS Terry Marshall Jr.

2. In Our Own Words • Using relationships among a statements and their converse, inverse, and contrapositive. • We need to be able to write and recognize converse ,inverse, and contrapostive.

3. Statement: a sentence that is either true or false, but not both • Example that is a statement: A triangle has three sides. • Example that is not a statement: Close the door.

4. Conditional Statement: a logical statement with two parts usually in “if-then” form • It has two parts: the hypothesis (“if” part) and the conclusion (“then” part)

5. CHANGES

6. TRUE OR FALSE • For a conditional statement to be true, that means that whenever the hypothesis is true, the conclusion is true also, with no exceptions! So, if there is even ONE exception, then you say it’s false! If you can come up with an exception that would be called a counterexample. • **A good way to remember that converse means to switch is to think about “converse” shoes switching back and forth when you are walking.

7. Example: • If a polygon has four sides, then itis a quadrilateral. (true) • What is the hypothesis? A polygon has four sides What is the conclusion? It is a quadrilateral • Inverse(not): If a polygon does not have four sides, then it is not a quadrilateral. (true) • Converse(switch): If a polygon is a quadrilateral, then it has four sides. (true) • Contrapositive(switch and not): If a polygon is not a quadrilateral,then it does not have four sides. (true)

8. TRUE OR FALSE • Example: If it is a dog, then it is an animal. (true) • What is the hypothesis? It is a dog. What is the conclusion? It is a an animal. • Inverse(not): If it is not a dog, then it is not an animal. (false) • Counterexample: A cat is not a dog which makes the hypothesis true, but it is an animal which makes the conclusion false. • Converse(switch): If is an animal, then it is a dog. (false) • Counterexample: A cat is an animal which makes the hypothesis true, but it is not a dog which makes the conclusion false. • Contrapositive(switch and not): If it is not an animal, then it is not a dog. (true)

9. TRUE AND FALSE • When two statements are both true or both false, they are called equivalent statements.

10. DEFINITIONS • ***The contrapositive is always logically equivalent to the original statement- meaning that the conditional statement is true, the contrapositive must also be true. • A biconditional statement has the words “if and only if” instead of just “if”. • You may write a biconditional statement only if the statement and it’s converse are both true.

11. Example: • Statement: If a whole number is even then it divisible by two. (true) • Converse: If a whole number is divisible by two, then it is even. (true) • Biconditional: A whole number is even if and only if it is divisible by two. • Example: Statement: If you are in Mrs. Kesler’s class then you are a student. (true) • Converse: If you are a student then you are in Mrs. Kesler ’s class. (false) • Biconditional: I can’t write a biconditional because the converse is false – not all students are in Mrs. Kesler’s class.