Logic Puzzles Four friends (Al, Gary, Mike, & Roger) each have a favorite TV show. Oddly enough, while the four are close friends, their interest in TV shows is so different that none of the friends has the same favorite. In fact, each favorite show (Explorer, FBI, Family Tides, & Z-Files) airs on a different night (Sunday, Monday, Tuesday, & Wednesday) and at a different time (7pm, 8pm, 9pm or 10pm). Determine the name of each friend, the title of their favorite TV show, and the day of the week and time that the show airs.

1. The person who enjoyed the TV show, FBI, doesn’t watch it on Sunday night. The TV show, Explorer, aired at 7 pm, but it wasn’t Mike’s favorite show. 2. Family Tides wasn’t Al’s favorite TV show. 3. Roger’s favorite TV show aired at 10 pm, but not on Wednesday night. 4. The TV show that aired on Tuesday night is The Z-Files. 5. Mike doesn’t watch a TV show at 9 pm. Gary doesn’t watch TV on Sunday night. 6. The TV show, Family Tides, aired an hour after the show that Gary liked but an hour before the show that aired on Monday night.

Conditional Statements (p q) if p, then q To solve a logic puzzle, you used “if – then” statements, called conditional statements ex. If the person who watches FBI doesn’t watch it on Sunday night, then FBI does not air on Sunday night. This is called the hypothesis This is called the conclusion

Truth Value A conditional statement can have a truth value of true or false. To be true, every time the hypothesis is true, the conclusion must also be true. To be false, there need only be one counterexample. Ex. If a number is a whole number, then it is a counting number.

Converse Statements (q p) If q then p A converse statement switches the hypothesis and conclusion. Conditional: If a student goes to Rogers-Herr, then he/she is in middle school. Converse: If a student is in middle school, he/she goes to Rogers-Herr. Even if conditional statements are true, their converses are not necessarily true!

Negation Statements ~p (not p) A negation statement has the opposite truth value of the original statement. Example: Raleigh is the capital of NC. Truth value: True! Negation: Raleigh is not the capital of NC. Truth value: False!

Inverse Statements (~p ~q) if not p, then not q An inverse statement negates both the hypothesis and the conclusion. Conditional: If a student goes to Rogers-Herr, then he/she is in middle school. Inverse: If a student does not go to Rogers-Herr, then he/she is not in middle school. Even if conditional statements are true, their inverses are not necessarily true!

Contrapositives (~q ~p) If not q, then not p A contrapositive statement switches and negates the hypothesis and conclusion. Conditional: If a student goes to Rogers-Herr, then he/she is in middle school. Contrapositive: If a student is not in middle school, he/she does not go to Rogers-Herr.

Truth Values If a conditional statement is true, is the converse statement also true? If a conditional statement is true, is the inverse statement also true? If a conditional statement is true, is the contraspositive statement also true?

Biconditional Statements (p q) p if and only if q When a conditional statement and it’s converse are both true, then the statement is biconditional. Conditional: If lines are skew, then they are not coplanar. Converse: If lines are not coplanar, then they are skew. Both are true, so this is biconditional! Lines are skew if and only if they are not coplanar.

Warm Up 1.17 Write a converse, inverse, contrapositive, and biconditional statement for this conditional statement: If you are in Miss Cline’s Core 4 class, then you are enrolled in geometry. Determine the truth value of each statement.

Warm Up Review Conditional: If you are in Miss Cline’s Core 4 class, then you are enrolled in geometry. Converse: If you are enrolled in geometry, then you are in Miss Cline’s Core 4 class. Inverse: If you are not in Miss Cline’s Core 4, then you are not enrolled in geometry. Contrapositive: If you are not enrolled in geometry, then you are not in Miss Cline’s Core 4 class. Biconditional: You take geometry if and only if you are in Miss Cline’s Core 4 class. F F T T F

Warm Up What number goes in the red square? 2. What reasoning did you use to determine that? 3. What number goes in the yellow square? 4. What reasoning did you use to determine that?

2 Types of Reasoning When you solve a Sudoku puzzle, you use two types of reasoning to figure out where each number goes – 1. Deductive reasoning 2. Indirect reasoning

Deductive Reasoning (or Logical Reasoning) - the process of reasoning from given statements to a conclusion. If you are a Dreamseeker, then Miss Cline is your math teacher. If you are a Dreamseeker, what can you conclude?

The Law of Detachment If a conditional is true and its hypothesis is true, then it’s conclusion is true. If p →q is a true statement and p is true, then q is also true. Ex. A gardener knows that if it rains, the gardens will be watered. It is raining. What can be concluded?

More Logical Reasoning If you are a Dreamseeker, then Miss Cline is your math teacher. If Miss Cline is your math teacher, then you are blessed! If you are a Dreamseeker, what can you conclude?

The Law of Syllogism If p →q and q →r are true statements, then p →r is a true statement. Ex. If the circus is in town, then there are tents at the fairground. If there are tents at the fairground, then Paul set up the tents. If the circus is in town, what can be concluded?

Indirect Reasoning In indirect reasoning, all possibilities are considered and all but one are proven false.

Warm Up 1.19 If Katie goes to play rehearsal, she will not attend soccer practice. If she does not attend soccer practice, then she will not play in Tuesday’s game. Katie went to play rehearsal. What can you conclude?

Warm Up 1.20 Prove this statement is true using indirect reasoning: If two angles are supplementary, only one of them can be obtuse. Hint – start by saying that if two angles are supplementary then both if them CAN be obtuse. Then, prove that this is false!