Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Survey

Transcript

C ONDITIONAL S TATEMENTS PARTNER C HALLENGE Amy, Bob and Carla are in a band. One is the drummer, one is the guitarist, and one is the keyboard player. Use the clues to find the instrument that each plays. Instrument Drums Guitar Keyboard Amy Bob Carla PARTNER C HALLENGE Instrument Amy Bob Carla Drums Guitar Keyboard •Carla and the drummer wear differentcolored shirts. •The keyboard player is older than Bob. •Amy, the youngest band member, lives next door to the guitarist. Vocabulary Conditional Statement – an if-then statement Example: If you are not completely satisfied, then your money will be refunded. Vocabulary hypothesis– the part of the conditional statement following “if”. Example: If you are not completely satisfied, then your money will be refunded. Vocabulary conclusion– the part of the conditional statement following “then”. Example: If you are not completely satisfied, then your money will be refunded. Your Turn: Identify the hypothesis and the conclusion of the following conditional statement. If y – 3 = 5, then y = 8. Your Turn: Write the following sentence as a conditional statement. An integer that ends with a zero is divisible by 5. Your Turn: Write the following sentence as a conditional statement. A square had four congruent sides. Vocabulary Truth value– determining if the conditional statement is TRUE or FALSE. A conditional statement is TRUE if every time the hypothesis is true, the conclusion is also true. Example of a true conditional statement: If a figure is a square, then it has four congruent sides. Vocabulary Truth value– determining if the conditional statement is TRUE or FALSE. A conditional statement is FALSE if you can find just one counterexample for which the hypothesis is true and the conclusion is false. Example of a false conditional statement: If you go to Wallenpaupack Area HS, then you live in Hawley, PA. Your Turn: Determine the truth value of the following conditional statement. If you live in Philadelphia, then you live in Pennsylvania. Your Turn: Determine the truth value of the following conditional statement. If a quadrilateral has four right angles, then the quadrilateral is a square. Vocabulary Venn Diagram– a diagram made up of overlapping circles/ovals. A Venn Diagram can be useful in determining the truth value of a conditional statement. Venn Diagram to represent a TRUE conditional statement. If you are legally driving, then you are at least 16 years old. Legal Drivers 16 year olds Venn Diagram to represent a FALSE conditional statement. If you play the flute, then you are in the band. Flute Players Band Members Vocabulary Converse of a Conditional Statement– switch the hypothesis and the conclusion. Example: Conditional Statement: If 2 lines are not parallel and do not intersect, then they are skew. Converse: If 2 lines are skew, then they are not parallel and do not intersect. Conditional Statement: If 2 lines are not parallel and do not intersect, then they are skew. True or False Converse: If 2 lines are skew, then they are not parallel and do not intersect. True or False Conditional Statement: If a figure is a square, then it has four sides. True or False Converse: If a figure has four sides, then it is a square. True or False Symbolic Form Conditional Statement– if p, then q. pq Converse– if q, then p. qp H OMEWORK pg. 71: 1-31 odd, 43-48, 54-58. Logic and Sudoku 2 9 3 7 6 4 9 7 1 5 3 8 9 1 6 8 3 7 4 5 3 9 7 6 4 1 6 9 2 1 4 5 1 8 1 5 3 8 4 9 5 9 6 2 7 4 2 3 8 3 7 4 1 8 6 2 9 Vocabulary Biconditional Statement – a statement you get by connecting the conditional statement and its converse with the word “and”. You can also use the phrase “if and only if”. Can only be combined if the conditional statement and its converse are both true. Example of a Biconditional: Conditional Statement: If two angles have the same measure, then the angles are congruent. Converse: If two angles are congruent, then the angles have the same measure. Since both the conditional and converse statements are true… Biconditional: Two angles have the same measure if and only if the two angles are congruent. Symbolic Form Conditional Statement– if p, then q. pq Converse– if q, then p. qp Biconditional– p if and only if q pq What makes a Good Definition? Uses clearly understood terms Precise (don’t use words such as sort of, or some) Reversible (can be written as a biconditional) Is this a Good Definition? A right angle is an angle whose measure is 90. Good Definition Conditional: If an angle is a right angle, then it measures 90. Converse: If an angle measures 90, then it is a right angle. Biconditional: An angle is a right angle if and only if its measure is 90. Is this a Good Definition? An airplane is a vehicle that flies. Bad Definition Conditional: If its an airplane, then it’s a vehicle that flies. Converse: If it’s a vehicle that flies, then it is an airplane. Counterexample: A helicopter is a vehicle that flies. H OMEWORK pg. 78: 1-23, 27, 32-35. PARTNER C HALLENGE Alan, Ben, and Cal are seated as shown with their eyes closed. Diane places a hat on each of their heads from a box that contains 3 red hats and 2 blue hats. They open their eyes and look forward. Alan says, “I cannot deduce what color hat I’m wearing.” Hearing that, Ben says, “I cannot deduce what color hat I’m wearing either.” Cal then says, “I know what color I am wearing.” What color is Cal’s hat? How does Cal know the color of his hat? Alan Ben Cal Vocabulary Negation– having the opposite truth value. Example of Negation: Statement: I studied 4 hours. Negation: I did not study 4 hours. Statement: I do not like reading books. Negation: I like reading books. Symbolic Form ~ Vocabulary Inverse of a Conditional Statement– negation of both the hypothesis and the conclusion. Conditional Statement: If two angles have the same measure, then the angles are congruent. Inverse: If two angles do not have the same measure, then the angles are not congruent. Vocabulary Contrapositive of a Conditional Statement– switches the hypothesis and the conclusion and negates both. Conditional Statement: If a figure is a square, then it is a rectangle. Contrapositive: If a figure is not a rectangle, then it is not a square. Switch and negate both Symbolic Form Conditional Statement– if p, then q. pq Negation– not p ~p Inverse– If not p, then not q ~p ~q Contrapositive– If not q, then not p ~q ~p Conditional Statement pq If a person is old enough to vote, then he/she is at least 18 years old. Converse qp If a person is at least 18 years old, then he/she is old enough to vote. ~p ~q Inverse If a person is NOT old enough to vote, then he/she is NOT at least 18 years old. Contrapositive ~q ~p If a person is at NOT least 18 years old, then he/she is NOT old enough to vote. Biconditional Statement A person is old enough to vote, if and only if he/she is at least 18 years old. H OMEWORK pg. 267: 1-9, 2227, 33-35, 42-44 Logic and Sudoku 2 9 3 7 6 4 9 7 1 5 3 8 9 1 6 8 3 7 4 5 3 9 7 6 4 1 6 9 2 1 4 5 1 8 1 5 3 8 4 9 5 9 6 2 7 4 2 3 8 3 7 4 1 8 6 2 9 Possible Hat Combinations: Alan Ben If Benwould saw blue, hewhat new Alan know that he wearing was wearing he was if he red. saw Therefore, he did not see two blue hats in front of blue. him. Therefore, Cal must be wearing red. R R R R B B B R R B B R R B Cal R B R B R B R