Section 2.2 Analyze Conditional Statements

1. Section 2.2Analyze Conditional Statements

2. What is an if-then statement? If-then statements can be used to clarify statements that may seem confusing. These statements are logic statements. Logic statements are important in many different types of professions.

3. Examples: • If the sun shines, then the grass will grow. • If I live in NJ, then I live on the east coast. • If the month is January, then next month is February.

4. These if-then statements are called conditional statements or conditionals. Conditional Statement: A logical statement that has two parts. In general, these conditionals are written: If p, then q or p q. Where p is the hypothesis and q is the conclusion.

5. Let’s take a look back at our examples: • If the sun shines, then the grass will grow. • If I live in NJ, then I live on the east coast. • If the month is January, then next month is February.

6. Converse: Exchange the hypothesis and conclusion of the conditional. The converse of p q is q p. Conditional:If I live in NJ, then I live on the east coast. Converse:If I live on the east coast, then I live in NJ. True! False!

7. B C A D Write the converse of the following statement and decide if it is true or false: Conditional: If two angles are adjacent, then they have a common side. Converse: If two angles have a common side, then they are adjacent. FALSE!! CAD and BAD share a common side, but they are not adjacent angles.

8. The denial of a statement is called a negation. ~p represents “not p” • 3.) This is geometry. • This is not geometry. • 4.) Today is not Thursday. • Today is Thursday. • 1.) An angle is obtuse. • An angle is not obtuse. • 2.) A puppy is a dog. • A puppy is not a dog.

9. These are congruent, but not vertical. The inverse is FALSE! 50º 50º The inverse of a conditional can be formed by negating both the hypothesis and conclusion. ~p ~q If-then Statement: If two angles are vertical, then they are congruent. Inverse: If two angles are not vertical, then they are not congruent.

10. Contrapositive: can be formed by negating the hypothesis and conclusion of the converse of the given conditional. Whoa! What does that mean????!!! ~q ~p If-then statement: If two angles are vertical, then they are congruent. Contrapositive: If two angles are not congruent, then they are not vertical. Is the contrapositive of this statement true or false???

11. Quick Review If-then statement: If p, then q. Converse: If q, then p. Inverse: If ~p, then ~q. Contrapositive: If ~q, then ~p.

12. Let’s put it all together! If-then statement: If you live in Red Bank, then you live in New Jersey. Converse: If you live in New Jersey, then you live in Red Bank. Inverse: If you do not live in Red Bank, then you do not live in New Jersey. Contrapositive: If you do not live in New Jersey, then you do not live in Red Bank.

13. Equivalent Statements • A conditional statement and its contrapositive are either both true or false. • The converse and inverse are either both true or both false. • When two statements are both true or both false, they are called equivalent statements.

14. Biconditional Statements • When a conditional statement and its converse are both true, you can write them as a biconditional statement. • Biconditional Statement: A statement that contains the phrase “if and only if”.

15. Rewrite as Biconditional Statements 1.) Rewrite the definition of right angle as a biconditional statement. An angle is a right angle if and only if the measure of the angle is 90 degrees.