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
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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
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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
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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
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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
