Unit 2: Logic and Reasoning. start of unit

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1 Unit 2: Logic and Reasoning Prior Unit: Introduction to Geometry Next Unit: Transversals By the end of this unit I will be able to: Skill Self-Rating start of unit Date(s) covered Self-Rating end of unit Identify, analyze, and create conditional statements. (4.B) Identify and validate the converse of a conditional statement. (4.B) Identify and validate the contrapositive of a conditional statement. (4.B) Identify and validate the inverse of a conditional statement. (4.B) Connect a biconditional statement with a true conditional statement and its true converse. (4.B) Verify a conjecture is false using a counter-example. (4.C) Create and use a Venn Diagram to analyze a conditional statement. (4.B) Analyze the converse, contrapositive, and inverse of a conditional statement using verbal and symbolic forms. (4.B) Create a true conditional statement and its converse, contrapositive, and inverse. (4.B) By the end of this unit I will be able to correctly define and apply these terms or formulas Conditional Statement Biconditional Statement Converse Contrapositive Inverse Counter-Example Verbal Form Symbolic Form Conjecture Venn Diagram Negation Reminders: Homework due each Friday, Quiz or Test each Friday, and one CFA every 6 weeks (usually in week 5). 1

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6 Conditional Statements A conditional statement, symbolized by p q, is an if-then statement in which p is a hypothesis and q is a conclusion. The logical connector in a conditional statement is denoted by the symbol. The conditional is defined to be true unless a true hypothesis leads to a false conclusion. Create 3 conditional statements, written in verbal form ( if p, then q ) Create 3 conditional statements, written in symbolic form ( p q )

7 Inverse, Converse, and Contrapositive of a Conditional Statement Negate Switch Switch AND Negate Negation The negation of statement p is "not p." The negation of p is symbolized by "~p." The truth value of ~p is the opposite of the truth value of p. Solution: Since p is true, ~p must be false. p: The number 9 is odd. true ~p: The number 9 is not odd. false Let's look at some more examples of negation. Example 1: r: 7 < 5 false ~r: 7 5 true Example 2: a: The product of two negative numbers is a positive number. true ~a: The product of two negative numbers is not a positive number. false 7

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9 Create a conditional statement (do not use examples already used or discussed in class) then create the inverse, converse, and contrapositive of the original statement. Original Statement Verbal Form ( if p, then q ) Symbolic Form ( p q ) Inverse Converse Contrapositive 9

10 Counter-Example One counter-example disproves a conditional statement. For example, the conditional statement: if a polygon is a triangle, then it has 3 acute angles Can be shown to be false if there is even one triangle that has less than 3 acute angles. We know it is false since there exist both right triangles and obtuse triangles that have at least one angle that is not acute. For the conditional statements below find a counter-example to prove them false. You can use a drawing or written reasoning. 1. if a quadrilateral has 4 right angles, then it is a square 2. if two lines intersect, then they then are perpendicular to each other 3. if a is greater than b, then c multiplied by a is greater than c multiplied by b (a, b, and c are real numbers). 4. if a polygon is a trapezoid, then it has no right angles 5. Create your own conditional statement and disprove it with a counter-example. 10

11 Conditional Statements using a Venn Diagram We can also represent conditional statements using a Venn Diagram. For example, the Venn Diagram below can represent the conditional statement: if an animal is poodle, then it is a dog The Venn Diagram below represents the generic conditional statement: if p, then q, with p being the hypothesis and q being the conclusion. Create 3 true conditional statements and represent them using a Venn Diagram. 11

12 Biconditional For a statement to be biconditional it means we could switch the hypothesis and conclusion and the resulting statement would always be true. Example: if a polygon has exactly 3 sides, then the polygon is a triangle This conditional statement is true. If we switch the hypothesis and conclusion we get: if a polygon is a triangle, then the polygon has exactly 3 sides. This conditional statement is also true so the original statement is biconditional. Example: if a rhombus has 4 right angles, then the rhombus is a square. This conditional statement is true. If we switch the hypothesis and conclusion we get: if the rhombus is a square, then it has 4 right angles This conditional statement is also true so the original statement is biconditional. Non-Example: if a polygon is a square, then the polygon is a rectangle This conditional statement is true. If we switch the hypothesis and conclusion we get: if the polygon is a rectangle, then is it a square. This conditional statement is not true since we can find at least 1 counter-example, thus the original statement is not biconditional. On the next page, create 2 examples of biconditional statements and 2 examples where the original conditional statement is true but the switched statement is not true. For the non-examples, provide a counter-example that disproves the switched statement. 12

13 Example 1: Example 2: Non-Example 1: Non-Example 2: 13

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