1. Section 2.2 Analyze 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: 1. If the sun shines, then the grass will grow. 2. If I live in NJ, then I live on the east coast. 3. 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. q is the conclusion.

5. Let’s take a look back at our examples: 1. If the sun shines, then the grass will grow. 2. If I live in NJ, then I live on the east coast. 3. 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. 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. Converse: If I live on the east coast, then I live in NJ. True! False!

7. 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. A B C D   CAD and  BAD share a common side, but they are not adjacent angles. FALSE!!

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

9. 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. 50º These are congruent, but not vertical. The inverse is FALSE!

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. A conditional statement and its contrapositive are either both true or false. The converse and inverse are either both true or both 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. 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. 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”. 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.