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Geometry

Geometry. Day 10. Today’s Agenda. Inductive reasoning Counterexamples Conditional Statements Inverse Converse Contrapositive Truth Tables Conjunctions Disjunctions Biconditionals Venn diagrams. Standards:.

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Geometry

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  1. Geometry Day 10

  2. Today’s Agenda • Inductive reasoning • Counterexamples • Conditional Statements • Inverse • Converse • Contrapositive • Truth Tables • Conjunctions • Disjunctions • Biconditionals • Venn diagrams

  3. Standards: • Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. • Find the converse, inverse, and contrapositive of a statement. • Use truth tables to determine the truth values of propositional statements.

  4. Inductive Reasoning • Inductive reasoning is the process of using examples and observations to reach a conclusion. • Any time you use a pattern to predict what will come next, you are using inductive reasoning. • A conclusion based on inductive reasoning is called a conjecture. • Turn to p. 90 and complete Guided Practice problems 1-2.

  5. Counterexamples • A conjecture is either true all of the time, or it is false. • If we wish to demonstrate that a conjecture is true all the time, we need to prove it through deductive reasoning. • We will have more on deductive reasoning and the proof process later. But for now, know that we can never prove an idea by offering examples that support the idea. • However, it can be easy to demonstrate that a conjecture is false. We simply need to provide a counterexample. • P. 92, Guided Practice 4

  6. Practice • Complete #48 on page 95. • Now complete #50. • Careful! Inductive reasoning is only as good as our observations. If we encounter new data that contradicts our conjecture, we need to revise the conjecture.

  7. Intro to Logic • A statement is a sentence that is either true or false (its truth value). • Logically speaking, a statement is either true or false. What are the values of these statements? • The sun is hot. • The moon is made of cheese. • A triangle has three sides. • The area of a circle is 2πr. • Statements can be joined together in various ways to make new statements.

  8. Conditional Statements • A conditional (or propositional) statement has two parts: • A hypothesis (or condition, or premise) • A conclusion (or result) • Many conditional statements are in “If… then…” form. • Ex.: If it is raining outside, then I will get wet. • A conditional statement is made of two separate statements; each part has a truth value. But the overall statement has a separate truth value. What are the values of the following statements? • If today is Friday, then tomorrow is Saturday. • If the sun explodes, then we can live on the moon. • If a figure has four sides, then it is a square.

  9. Conditional Statements • Conditional statements don’t have to be “If… then…” See if you can determine the condition and conclusion in each of the following, and restate in “If… then…” form. • An apple a day keeps the doctor away. • What goes up must come down. • All dogs go to heaven. • Triangles have three sides.

  10. Inverse • The inverse of a statement is formed by negating both its premise and conclusion. • Statement: • If I take out my cell phone, then Mr. Peterson will confiscate it. • Inverse: • If I do take out my cell phone, then Mr. Peterson will confiscate it. not not

  11. Try these • Give the inverses for the following statements. (You may wish to rewrite as “If… then…” first.) Then determine the truth value of the inverse. • Barking dogs give me a headache. • If lines are parallel, they will not intersect. • I can use the Pythagorean Theorem on right triangles. • A square is a four-sided figure.

  12. Converse • A statement’s converse will switch its hypothesis and conclusion. • Statement: • If I am happy, then I smile. • Converse: • If , then . I am happy I smile

  13. Try these • Give the converses for the following statements. Then determine the truth value of the converse. • If I am a horse, then I have four legs. • When I’m thirsty, I drink water. • All rectangles have four right angles. • If a triangle is isosceles, then two of its sides are the same.

  14. Contrapositive • A contrapositive is a combination of a converse and an inverse. The premise and conclusion switch, and both are negated. • Statement: • If my alarm has gone off,then I am awake. • Contrapositive: • If ,then . my alarm has not gone off not I am not awake not

  15. Try these • Give the contrapositives for the following statements. Then determine its truth value. • If it quacks, then it is a duck. • When Superman touches kryptonite, he gets sick. • If two figures are congruent, they have the same shape and size. • A pentagon has five sides. • Note: A contrapositive always has the same truth value as the original statement!

  16. Symbolic representation • Logic is an area of study, related to math (and computer science and other fields). In formal logic, we can represent statements symbolically (using symbols). • Some common symbols: a statement, usually a premise a statement, usually a conclusion creates a conditional statement negates a statement (takes its opposite)

  17. Examples • If p, then q • Inverse:If not p, then not q • Converse:If q, then p • ContrapositiveIf not q, then not p

  18. Truth Table • A truth table is a way to organize the truth values of various statements. • In a truth table, the columns are statements and the rows are possible scenarios. • The table contains every possible scenario and the truth values that would occur. • Example: T F F T

  19. A conditional truth table T T T T F F F T T F F T

  20. A conditional truth table T T T T T T T F F T F T F T F F T T F F T T T T

  21. Logical Equivalents • Two statements are considered logical equivalents if they have the same truth value in all scenarios. A way to determine this is if all the values are the same in every row in a truth table.

  22. Logical Equivalents • Which of the following statements are logically equivalent? T T T T T T T F F T F T F T F F T T F F T T T T

  23. Conjunctions • A conjunction consists of two statements connected by ‘and’. • Example: • Water is wet and the sky is blue. • Notation: • A conjunction of p and q is written as

  24. Conjunctions • A conjunction is true only if both statements are true. • Remember: the truth value of a conjunction refers to the statement as a whole. • Consider: “The sun is out and it is raining.” T T T T F F F T F F F F

  25. Disjunctions • A disjunction consists of two statements connected by ‘or’. • Example: • I can study or I can watch TV. • Notation: • A disjunction of p and q is written as

  26. Disjunctions • A disjunction is true if either statement is true. • Consider: “Timmy goes to Stanton or he goes to Paxon.” T T T T F T F T T F F F

  27. Biconditional • A biconditional statement is a special type of conditional statement. It is formed by the conjunction of a statement and its converse. • Example: • If a quadrilateral has four right angles then it is a rectangle, and if a quadrilateral is a rectangle then it has four right angles. • Biconditional statements can be shortened by using “if and only if” (iff.). • A quadrilateral is a rectangle if and only if it has four right angles. • This is true whether you read it forwards or ‘backwards’.

  28. Biconditional • A good definition will consist of a biconditional statement. • Ex: A figure is a triangle if and only if it has three sides.

  29. Biconditional • A biconditional is true when the statements have the same truth value. • Consider: “Two distinct coplanar lines are parallel if and only if they have the same slope.” • “Our team will win the playoffs if and only if pigs fly.” T T T T F F F T F F F T

  30. Venn Diagrams • The truth values of compound statements can also be represented in Venn diagrams. • p: A figure is a quadrilateral. • q: A figure is convex. • Which part of the diagramrepresents: p q

  31. Venn Diagrams – Conditionals • A Venn diagram can represent a conditional statement: • p: A figure is a quadrilateral. • q: A figure is a square. p q

  32. Can you…? • Use inductive reasoning to recognize patterns and make predictions? • Give a counter-example to disprove a conjecture? • Identify the hypothesis and conclusion of a conditional statement? • Write the converse, inverse, and contrapositive of a conditional statement? • Create a truth table to examine scenarios? • Recognize conjunctions, disjunctions, and biconditional statements? • Evaluate logic using Venn diagrams?

  33. Assignments • Homework 5 • Wkbk, pp. 15, 19 • Homework 6 • Truth Tables Handout • Textbook • pp. 102-103; #31, 33, 41-47 • pp. 112; #59-61

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