Understanding Conditional Statements and Logical Equivalences

2. 2.1 Conditional Statements

3. Conditional Statement – a logical statement with two parts. P, the “if part”, hypothesis Q, the “then part”, conclusion If p, then q is symbolized as p -> q (p implies q). Ex: If you study hard, then you will get As. If you live in Ridgewood, then you live in NJ. If an angle is 28 degrees, then it is acute.

4. Rewrite into “If, then” form • All fruits have seeds. • Two angles that add up to 180 degrees are supplementary. • An even number is a number divisible by 2. • 3x + 17 = 23, because x = 2. • Only people who are 18 are allowed to vote. • P=It is raining. Q =I must bring an umbrella. • Today is Friday and tomorrow is Saturday. Now go back and underline the hypothesis and circle the conclusions.

5. Negation ~ The negation of a statement is the opposite of the original statement. Symbol: ~p Words: not p Examples: Find the negation of the statement. • This suit is black. • We are worthy.

6. Converse, Inverse and Contrapostive Given Conditional Statement: p->q Converse: Switch hypothesis and conclusion. q->p Inverse: Negate hypothesis and conclusion. ~p->~q Contrapositive: Switch and negate the hypothesis and conclusion. ~q->~p

7. Counterexamples A counterexample is a case that fits the hypothesis but leads to a different conclusion. You can prove a statement is false by finding a counterexample. Practice finding counterexamples. 1) If a number is prime, then it is odd. 2) If you live in Ridgewood, you go to RHS. 3) If a food is round, then it is pizza. 4) If a 3D figure lacks vertices, it is a sphere.

8. Conditional Statement: If you are a guitar player, then you are a musician. Converse: If you are a musician, then you are a guitar player. T or F Inverse: If you are not a guitar player, then you are not a musician. T or F Contrapositive. If you are not a musician, then you are not a guitar player. T or F

9. CONnie Mack was a SWITCH Hitter Associate CONverse with SWITCHING the hypothesis and conclusion

10. CONverses help me do a FLIP

11. The CONvictSWITCHES his clothes once he escapes from jail.

12. N*Sync has a song NO strings attached No Strings Attached

13. INverse think NOT

14. INdigo is not blue or not purple

15. If you are INdignant, you are NEGATIVE

16. Contrapositive • You SWITCH and NEGATE the hypothesis and conclusion. • It is like taking the converse and inverse at the same time.

17. Fun Fact • A conditional statement and its contrapositive are either both true or false. They are equivalent statements. • A converse statement and an inverse statement are also both true or false. They are also equivalent statements. • But (for example), a conditional may be true and a converse may be false.

18. Guided Practice Everyone will have to share, so make sure you come up with something. • Write the definition of perpendicular lines as a conditional statement. Is the converse true? • Come up with your own conditional statement. Then find the converse, inverse and contrapositive. • Write true or false next to each of the four statements you wrote in number 2.

19. Biconditional Statements When a conditional statement and its converse are true, you can write them as a single biconditional statement linking the hypothesis and conclusion with “if and only if”. Words: p if and only if q Symbol: p <-> q Write a biconditional statement for your conditional statement if both your conditional and converse were true. If not, write one for perpendicular lines.