Warm Up Underline the hypothesis and circle the conclusion of each conditional. 1. A mapping that is a reflection is a type of transformation. 2. The quotient of two negative numbers is positive. 3. Determine if the conditional “If x is a number then |x| > 0” is true. If false, give a counterexample. 2.3 Logic Laws

Objective Apply the Law of Detachment and the Law of Syllogism in logical reasoning. 2.3 Logic Laws Deductive reasoning is the process of using __________to draw conclusions from given ________, ________________, and ___________________. logic facts definitions properties Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true.

Example 1 Decide whether inductive or deductive reasoning is used to reach the conclusion. Explain your reasoning. a.At Reagan High School, students must pass Geometry before they take Algebra 2. Emily is in Algebra 2, so she must have passed Geometry. b.Sally is taking the bus to school. She notices that the next 5 people who get on the bus are wearing jeans. She concludes that the next person to get on the bus will also be wearing jeans. c.Kim knows she is younger than Jeff, but she is older than Phil. So Kim concludes that Jeff must be older than Phil.

Law of Detachment If p  q is a true statement and p is true, then q is true. There are two laws of deductive reasoning.

Example 2 Use the Law of Detachment to determine the conclusion. If a car is a corvette, then it is a Chevrolet. Suzie's car is a Corvette. Conclusion Suzie's car is a Chevrolet.

Example 3 Determine if the conjecture is valid by the Law of Detachment. Given: In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the 2004 World Series. Conjecture: The Red Sox won the 2004 World Series. In the World Series, if a team wins four games, then the team wins the series. The Red Sox won four games in the 2004 World Series hypothesis conclusion hypothesis conclusion The Red Sox won the 2004 World Series. The conjecture is valid.

If a student passes his classes, then the student is eligible to play sports. Example 4 Determine if the conjecture is valid by the Law of Detachment. Given: If a student passes his classes, the student is eligible to play sports. Ramon passed his classes. Conjecture: Ramon is eligible to play sports. The statement “Ramon passed his classes” matches the hypothesis of a true conditional. By the Law of Detachment, Ramon is eligible to play sports. The conjecture is valid.

Example 5 Determine if the conjecture is valid by the Law of Detachment. Given: If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area of the triangle is 30 cm 2. The area of ∆PQR is 30 cm 2. Conjecture: The side lengths of ∆PQR are 5cm, 12 cm, and 13 cm. If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area of the triangle is 30 cm 2. The given statement “The area of ∆PQR is 30 cm 2 ” matches the conclusion of a true conditional. But this does not mean the hypothesis is true. The dimensions of the triangle could be different. So the conjecture is not valid.

Law of Syllogism If p  q and q  r are true statements, then p  r is a true statement.

Example 6 Determine if the conjecture is valid by the Law of Syllogism. Given: If a figure is a kite, then it is a quadrilateral. If a figure is a quadrilateral, then it is a polygon. Conjecture: If a figure is a kite, then it is a polygon.

Example 7 Determine if the conjecture is valid by the Law of Syllogism. Given: If a number is divisible by 2, then it is even. If a number is even, then it is an integer. Conjecture: If a number is an integer, then it is divisible by 2.

Example 8 Put the following statements in order. Given: If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal. Conclusion? __________________________________________________ __________________________________________________ Law of logic? _________________________________________________ If an animal is a mammal, then it has hair. If an animal is a dog, then it is a mammal.

Example 9 Put the following statements in order. Then use the Law of Syllogism to draw a conclusion. a. If quompies plaun, then romples gleer. b. If homblers frain, then quompies plaun. c. If ruskers bleer, then homblers frain. a. If quompies plaun, then romples gleer. b. If homblers frain, then quompies plaun. c. If ruskers bleer, then homblers frain. Conclusion? ________________________________________ __________________________________________________

Example 10 Draw a conclusion from the given information. a.Given: If 2y = 4, then z = –1. If x + 3 = 12, then 2y = 4. x + 3 = 12 Conclusion? ______________________________________________________________________ b. Given: If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle. Conclusion? _______________________________________________________________________

c. Given: If n is a natural number, then n is an integer. If n is an integer, then n is a rational number is a rational number. Conclusion? _______________________________________________________ ________________________