The document discusses conditional and biconditional statements in logic. It defines conditional statements using "if...then" and biconditional statements using "if and only if". It also discusses the converse, inverse, and contrapositive of conditional statements and how their truth values relate using truth tables. Specifically, the contrapositive of a conditional statement always has the same truth value as the original conditional statement.

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

2. 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? Introduction Instruction Examples Practice

3. Please go back or choose a topic from above. Introduction Instruction Examples Practice

4. List of Instructional Pages Two New Connectives The Conditional The Conditional – Truth Table The Biconditional The Biconditional – Truth Table Other If…then Statements The Converse 8. The Inverse 9. The Contrapositive 10. Conditional/ Converse – Truth Tables 11. Conditional/ Inverse – Truth Tables 12. Conditional/ Contrapositive – Truth Tables 13. Summary

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

6. This is page 2 of 13 Page list Last Next The Connectives: The Conditional A conditional expresses the notion of if . . . then . We use an arrow,  , to represent a conditional. p : You will study hard. s : You will get a good score in the exam. p  s : If you will study hard, then you will get a good score in the exam. “ If you will not study hard, then you will not get a good score in the exam.” would be written as ~p  ~s. Introduction Instruction Examples Practice

7. 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 Last Next Not exactly the same thing in Geometry! If..then statements related to conditionals Do they all mean the same thing? Introduction Instruction Examples Practice

8. Give the converse, inverse and contrapositive of the given conditional. Conditional: p  q “if p, then q” “ If you are a Filipino, then you are Asian.” Converse: q  p “if q, then p” “ If you are Asian, then you are a Filipino.” Inverse: ~p  ~q “if not p, then not q” “ If you are not a Filipino, then you are not Asian.” Contrapositive: ~q  ~p “if not q, then not p” “ If you are not Asian, then you are not a Filipino.”

9. Give the converse, inverse and contrapositive of the given conditional. Conditional: p  q “ If the figure is a square, then it has four sides.” Converse: q  p “ If the figure has four sides, then it is a square.” Inverse: ~p  ~q “ If the figure is not a square, then it doesn’t have four sides.” Contrapositive: ~q  ~p “ If the figure doesn’t have four sides, then it is not a square”

10. Give the converse, inverse and contrapositive of the given conditional. Conditional: “ If the animal has wings, then it is a bird.” Converse: “ If the animal is a bird, then it has wings.” Inverse: “ If the animal has no wings, then it is not a bird.” Contrapositive: “ If the animal is not a bird, then it does not have wings”

11. Give the converse, inverse and contrapositive of the given conditional. Conditional: “ If the animal has feathers, then it is a bird.” Converse: “ If the animal is a bird, then it has feathers.” Inverse: “ If the animal has no feathers, then it is not a bird.” Contrapositive: “ If the animal is not a bird, then it does not have feathers.”

12. Give the converse, inverse and contrapositive of the given conditionals. a : If today is Sunday, then it is a weekend day. b : If the figure has three sides, then it is a triangle. c : If Charlie is a basketball player, then Charlie is tall. d : If you are a teenager, then you are 13 years old.

13. 2. The car is washed but the ₱ 10 was not paid. The promise is not kept so the conditional is false. The car is not washed and the ₱ 10 is not paid. The promise is not broken since the car was not washed, so the conditional is still true. If the car was not washed, the payment does not have to be either true or false. Either way, the promise is kept, so the conditional is true. The car is washed and the ₱ 10 is paid. The promise is kept so the conditional is true. 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 wash my car, then I will pay you ₱ 10.” (p  q) There are four situations possible. This is page 3 of 13 Exercise Last Next The Conditional Introduction Instruction Examples Practice p q p  q T T T T F F F T T F F T

14. Let’s say p represents the statement “Marge lives in Cebu,” and q represents the statement “Marge lives in the Philippines.” This is page 7 of 13 Page list Last Next p  q is “If Marge lives in Cebu , then she lives in the Philippines .” The Converse q  p “ If Marge lives in the Philippines , then she lives in Cebu .” Just because the conditional is true does not mean the converse is true. TRUE Converse FALSE Introduction Instruction Examples Practice

15. Let’s look at the inverse. This is page 8 of 13 Page list Last Next p  q is “ If Marge lives in Cebu , then she lives in the Philippines .” The Inverse ~p  ~q “ If Marge does not live in Cebu , then Marge does not live in the Philippines Marge could still live in the Philippines and not be in Cebu. Just because the conditional is true does not mean the inverse is true. Inverse TRUE FALSE Introduction Instruction Examples Practice

16. Let’s look at the contrapositive. This is page 9 of 13 Page list Last Next p  q is “If Marge lives in Cebu , then she lives in the Philippines .” The Contrapositive ~q  ~p “If Marge does not live in the Philippines , then she does not live in Cebu .” If Marge isn’t in the Philippines, she can’t be in Cebu. If the conditional is true then the contrapositive is also true. Contrapositive TRUE TRUE Introduction Instruction Examples Practice

17. Slides after these are for optional study.

18. A biconditional expresses the notion of if and only if . Its symbol is a double arrow,  . p : If a polygon has three sides then it is a triangle . t : If a figure is a triangle then it is a polygon with three sides. p  t : “A polygon has three sides if and only if it is a triangle.” This is page 4 of 13 Page list Last Next The Connectives – the Biconditional Introduction Instruction Examples Practice

19. This is page 5 of 13 Page list Last Next The Biconditional 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 ” 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 Introduction Instruction Examples Practice p q p  q q  p (p  q)  ( q  p) (or p  q) T T T T T T F F T F F T T F F F F T T T

20. Let’s compare the truth tables for the conditional and the converse. This is page 10 of 13 Page list Last Next Comparing Truth Tables Confirms Our Conjectures The Converse The Conditional These two truth tables are not the same so the statements are not logically equivalent. Introduction Instruction Examples Practice p q p  q T T T T F F F T T F F T p q q  p T T T T F T F T F F F T

21. Let’s compare the truth tables for the conditional and the inverse. This is page 11 of 13 Page list Last Next Comparing Truth Tables Confirms Our Conjectures The Inverse The Conditional These two truth tables are not the same so the statements are not logically equivalent. Introduction Instruction Examples Practice p q p  q T T T T F F F T T F F T p q ~p ~q ~p  ~q T T F F T T F F T T F T T F F F F T T T

22. Let’s compare the truth tables for the conditional and the contrapositive. This is page 12 of 13 Page list Last Next Comparing Truth Tables Confirms Our Conjectures The Contrapositive These two truth tables are the same so the statements are logically equivalent. The Conditional Introduction Instruction Examples Practice p q ~q ~p ~q  ~p T T F F T T F T F F F T F T T F F T T T p q p  q T T T T F F F T T F F T

23. Let’s summarize the relationships: Conditional and Contrapositive always has the same truth value. Converse and Inverse always has the same truth value. This is page 13 of 13 Page list Last Next Summary p  q  q  p p  q  ~p  ~q p  q  ~q  ~p Converse Inverse Contrapositive q  p  ~p  ~q Introduction Instruction Examples Practice

24. Please go back or choose a topic from above. Introduction Instruction Examples Practice

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

26. Please go back or choose a topic from above. Introduction Instruction Examples Practice

27. Practice Gizmos: Conditional Statement Biconditional Statement How can we tell if a conditional statement is true or false? Introduction Instruction Examples Practice

28. Please go back or choose a topic from above. Introduction Instruction Examples Practice

29. 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.

30. Example 2 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 : 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 : If the football game was not cancelled, then it did not rain all day Friday. True .

31. 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 Class attendance will be down if the surf is up. If class attendance is down, then the surf is up. Converse, NAT