Inductive Reasoning and Conditional Statements

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Presentation transcript:

Inductive Reasoning and Conditional Statements

Inductive Reasoning - reasoning based upon patterns Monday, Tuesday, _________

Example: a) A gardener knows that if it rains, the garden will be watered. If it is raining, what conclusion can be drawn?

Example: b) My dog Danger runs in circles when he has to go to the bathroom. If Danger is running in circles, then what conclusion can I draw?

Conditional statement - If-then statements Always have two parts If I study my geometry notes, then I will pass my geometry test. Hypothesis – follows if Conclusion – follows then

Example: Identify the hypothesis and conclusion. a) If x + 3 = 12, then x = 9. b) If two lines are parallel, then they are coplanar. H: x + 3 = 12 C: x = 9 H: two lines are parallel C: they are coplanar

Example: Identify the hypothesis and conclusion. This time like all times is a very good one if we but know what to do with it ~ Ralph Waldo Emerson

Example: Write the statement as a conditional. a) An obtuse angle measures more than 90o If an angle is obtuse, then it measures more than 90o. If an angle measures more than 90o, then it is obtuse.

Example: Write the statement as a conditional. b) a square has four congruent sides If a figure is a square, then it has four congruent sides. If a figure has four congruent sides, then it is a square.

Try It Yourself! Hot Topics book p. 263 #1-6

Truth – Value - Is the statement true or false? IF FALSE: then you need to find a counterexample

Example: Determine the truth value. If false, give a counterexample. a) If x2 > 0, then x > 0.

Example: Determine the truth value. If false, give a counterexample. b) If a state’s name contains the word ‘new’, then it borders an ocean.

Example: Determine the truth value. If false, give a counterexample. c) If you are a girl, then you love to go shopping.

Inductive Reasoning and Conditional Statement Worksheet Homework Inductive Reasoning and Conditional Statement Worksheet

Venn Diagram Flight less birds penguins

Conditional Notation If you do your work, then you will pass Geometry. p  q p: hypothesis: you do your work q: conclusion: you will pass Geometry

Converse - Switch the hypothesis and conclusion original p  q: converse: q  p If you are a male, then you love football. If you love football, then you are a male.

Try It Yourself! Hot Topics book p. 263 #7-10

Inverse - negate the hypothesis and conclusion original p  q: inverse: ~p  ~q If you love math, then you are a math dork. If you do not love math, then you are not a math dork.

Try It Yourself! Hot Topics book p. 263 #11-16

Contrapositive - Switch the hypothesis and conclusion and negate original p  q: contrapositive: ~q  ~p If you love math, then you are a math dork. If you are not a math dork, then you do not love math.

Try It Yourself! Hot Topics book p. 263 #17-18

Example: Write the converse, inverse, and contrapositive of each statement. a) If two lines do not intersect, then they are parallel. If two lines are parallel, then the lines are do not intersect. If two lines intersect, then the lines are not parallel. If two lines are not parallel, then the two lines intersect.

Example: Write the converse, inverse, and contrapositive of each statement. b) If x=2, then lxl = 2. If lxl =2, then x=2. If x≠2, then lxl≠2. If lxl≠2, then x≠2.

Example: Write the converse, inverse, and contrapositive of each statement. c) If you could choose one characteristic that would get you through life, choose a sense of humor. ~ Jennifer Jones If you chose to not have a sense of humor, then you choose to not have the one characteristic that will get you through life. If you choose to have a sense of humor, then you chose the one characteristic that will get you through life. If you do not choose one characteristic that would get you through life, then you would not choose a sense of humor.

Try It Yourself! Hot Topics book p. 263 #20

Biconditional - When a conditional and its converse are true, you can combine them with ‘if and only if’ or ‘iff’

Example: Write a biconditional if possible. a) If x = 5, then x + 15 = 20. converse: If x + 15 = 20, then x =5. biconditional: x + 15 = 20 iff x = 5

Example: Write a biconditional if possible. b) If three points are collinear, then they lie on the same line. converse: If three points lie on the same line, then the points are collinear. biconditional: Three points are collinear iff they lie on the same line.

Example: Write a biconditional if possible. c) If a figure is a square, then it has four right angles. converse: If a figure has four right angles, then it is a square. biconditional: A figure is a square iff it has 4 right angles.

Try It Yourself! Workbook p. 18 #1-4

Example: Write the biconditional as two statements. a) Lines are skew iff they are noncoplanar. If lines are skew, then they are noncoplanar. If lines are noncoplanar, then they are skew.

Put All of This Stuff in Your Study Homework Put All of This Stuff in Your Study Guide!