1. Logic The Conditional and Related Statements Resources: HRW Geometry, Lesson 12.3

2. IntroductionInstructionExamplesPractice Logic and Geometry are both about developing good arguments or proofs that something is true or false. The other two connectives that create compound statements in logic, the conditional statement, and the biconditional statement are often involved in arguments and proofs. How can we tell if a conditional statement is true or false?

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4. List of Instructional Pages 1.Two New ConnectivesTwo New Connectives 2.The ConditionalThe Conditional 3.The Conditional – Truth TableThe Conditional – Truth Table 4.The BiconditionalThe Biconditional 5.The Biconditional – Truth TableThe Biconditional – Truth Table 6.Other If…then StatementsOther If…then Statements 7.The ConverseThe Converse 8. The InverseThe Inverse 9. The ContrapositiveThe Contrapositive 10. Conditional/ Converse – Truth TablesConditional/ Converse – Truth Tables 11.Conditional/ Inverse – Truth TablesConditional/ Inverse – Truth Tables 12.Conditional/ Contrapositive – Truth TablesConditional/ Contrapositive – Truth Tables 13. SummarySummary

5. IntroductionInstructionExamplesPractice We are ready to add the conditional and biconditional to our list of connectives. This is page 1 of 13 Page list LastNext Negation:NOTConjunction:AND Disjunction:OR Conditional:if…then Biconditional: if and only if The Symbols: NOT~ AND  OR  If…then  If and only if  The Connectives - Conditional and Biconditional

6. IntroductionInstructionExamplesPractice This is page 2 of 13 Graphic Page list LastNext The Connectives: The Conditional conditional , A conditional expresses the notion of if... then. We use an arrow, , to represent a conditional. p : The Braves will win the World Series. s : Susan will win an Emmy.  p  s : If the Braves win the World Series, then Susan will win an Emmy.  “If the Braves do not win the World Series, then Susan will not win an Emmy” would be written symbolically as ~p  ~s

7. 2.The lawn is mowed and the $10 is not paid. The promise is not kept so the conditional is false. 4.The lawn is not mowed and the $10 is not paid. The promise is not broken since the lawn was not mowed, so the conditional is still true. 3.The lawn is not mowed and the $10 is paid. The promise is kept, so the conditional is true. 1.The lawn is mowed and the $10 is paid. The promise is kept so the conditional is true. IntroductionInstructionExamplesPractice A conditional statement uses the words if…then. It is like making a promise. In logic, if the “promise” is broken, and not kept, the conditional is said to be false. Otherwise, it is true. Consider the statement: “If you mow my lawn, then I will pay you $10.” (p  q) There are four situations possible. This is page 3 of 13 Page list LastNext The Conditional pq p  q TTT TFF FTT FFT

8. IntroductionInstructionExamplesPractice biconditional . A biconditional expresses the notion of if and only if. Its symbol is a double arrow, . p : A polygon has three sides. t : The polygon is a triangle.  p  t : “A polygon has three sides if and only if it is a triangle” This is page 4 of 13 Page list LastNext The Connectives – the Biconditional

9. IntroductionInstructionExamplesPractice This is page 5 of 13 Page list LastNext The Biconditional pq p  q q  p (p q)  (q  p) (or p  q) TTTTT TFFTF FTTFF FFTTT  A biconditional (p  t) is a more concise way to say (p  t)  (t  p). If a polygon has three sides then it is a triangle If a figure is a triangle then it is a polygon with three sides “If a polygon has three sides then it is a triangle” and “If a figure is a triangle then it is a polygon with three sides” are both true statements. T A biconditional is true when both p  q and q  p are true. T

10. IntroductionInstructionExamplesPractice There are three other if…then statements related to a conditional statement, p  q. They are called: Converse: q  p Inverse: ~p  ~q Contrapositive: ~q  ~p This is page 6 of 13 Page list LastNext Not exactly the same thing in Geometry! If..then statements related to conditionals Do they all mean the same thing?

11. IntroductionInstructionExamplesPractice Let’s say p represents the statement “Marge lives in South Carolina,” and q represents the statement “Marge lives in the United States.” This is page 7 of 13 Page list LastNext p  q is “If Marge lives in South Carolina, then she lives in the United States.” The Converse q  p “If Marge lives in the United States, then she lives in South Carolina.” Just because the conditional is true does not mean the converse is true. TRUE Converse FALSE

12. IntroductionInstructionExamplesPractice Let’s look at the inverse. This is page 8 of 13 Page list LastNext p  q is “If Marge lives in South Carolina, then she lives in the United States.” The Inverse ~p  ~q “If Marge does not live in South Carolina, then Marge does not live in the United States.” Marge could still live in the US and not be in SC. Just because the conditional is true does not mean the inverse is true. Inverse TRUE FALSE

13. IntroductionInstructionExamplesPractice Let’s look at the contrapositive. This is page 9 of 13 Page list LastNext p  q is “If Marge lives in South Carolina, then she lives in the United States.” The Contrapositive ~q  ~p “If Marge does not live in the United States, then she does not live in South Carolina.” If Marge isn’t in the US she can’t be in SC. If the conditional is true then the contrapositive is also true. Contrapositive TRUE TRUE

14. IntroductionInstructionExamplesPractice Let’s compare the truth tables for the conditional and the converse. This is page 10 of 13 Page list LastNext Comparing Truth Tables Confirms Our Conjecturespq p  q TTT TFF FTT FFT pq q  p TTT TFT FTF FFT The Converse The Conditional These two truth tables are not the same so the statements are not logically equivalent.

15. IntroductionInstructionExamplesPractice Let’s compare the truth tables for the conditional and the inverse. This is page 11 of 13 Page list LastNext Comparing Truth Tables Confirms Our Conjecturespq p  q TTT TFF FTT FFT pq~p~q ~p  ~q TTFFT TFFTT FTTFF FFTTT The Inverse The Conditional These two truth tables are not the same so the statements are not logically equivalent.

16. IntroductionInstructionExamplesPractice Let’s compare the truth tables for the conditional and the contrapositive. This is page 12 of 13 Page list LastNext Comparing Truth Tables Confirms Our Conjectures pq~q~p ~q  ~p TTFFT TFTFF FTFTT FFTTT The Contrapositive These two truth tables are the same so the statements are logically equivalent.pq p  q TTT TFF FTT FFT The Conditional

17. IntroductionInstructionExamplesPractice Let’s summarize the relationship between a conditional statement and its converse, inverse, and contrapositive. This is page 13 of 13 Page list LastNext Summary p  q  q  p p  q  ~p  ~q p  q  ~q  ~p Converse Inverse Contrapositive

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19. IntroductionInstructionExamplesPractice Example 1 Example 2 Example 3 Examples 1.Writing “if…then” statements 2.Writing the converse, inverse, or contrapositive of a conditional statement. 3.Recognizing the converse, inverse, contrapositive given a conditional statement. IF THEN

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21. IntroductionInstructionExamplesPractice Gizmos: Conditional Statement Biconditional Statement How can we tell if a conditional statement is true or false?

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23. Example 1 All freshmen should report to the cafeteria. Back to main example page Rewrite each statement in if…then form. For example: “Every triangle is a polygon” becomes “If a figure is a triangle, then the figure is a polygon.” If you are a freshman, then you should report to the cafeteria. Reading horror stories at bedtime gives me nightmares. If I read a horror story at bedtime, then I will have nightmares. Driving too fast often results in accidents. If you drive too fast, then you are likely to have an accident.

24. Example 2 Converse Converse: If the football game was cancelled, then it must have rained all day Friday. Back to main example page Write the converse, inverse, and contrapositive for the given conditional statement. Decide whether each is true or false and explain your reasoning. “If it rains all day Friday, then the football game will be cancelled.” False. The game could have been cancelled because of something else, like a bomb threat. Inverse Inverse: If it did not rain all day Friday, then the football game was not cancelled. False. Just because it didn’t rain doesn’t mean the game couldn’t be cancelled for another reason. Contrapositive Contrapositive: If the football game was not cancelled, then it did not rain all day Friday. True.

25. Example 3 If I read the book, then I can do the homework. If I cannot do the homework, then I did not read the book. Back to main example page For each statement, name the relationship (converse, inverse, contrapositive) of the second statement to the first. State whether the second is always true (AT) or not always true (NAT) assuming pq is true. Contrapositive, AT If it is Tuesday, I go to geometry. If I go to geometry, it is Tuesday Converse, NAT If it is snowing, then it is cold. If it isn’t snowing, then it isn’t cold. Inverse, NAT if Class attendance will be down if the surf is up. If class attendance is down, then the surf is up. Converse, NAT