Consider the following TRUE conditional statement… UNIT 2 Deductive Reasoning 2.3 Consider the following TRUE conditional statement… If you miss the bus, then you will be late for school. Tony missed the bus. What can you conclude? Gina was late for school. What can you conclude?

UNIT 2 Deductive Reasoning 2.3 You only need one counter example to show a statement is false. To prove the statement is TRUE using inductive logic, you must show it is true for every possible case! Mathematicians turn to deductive reasoning to prove conjectures are true. Deductive logic uses facts, definitions and properties to draw conclusions.

If p  q is a true statement and p is true, then q is true. UNIT 2 Deductive Reasoning 2.3 The Law of Detachment is one valid form of deductive reasoning. Law of Detachment If p  q is a true statement and p is true, then q is true. Consider the conditional statement: If you miss the bus, then you will be late for school. Tony missed the bus. Gina was late for school.

If you like donkers, then you will like torks. UNIT 2 Deductive Reasoning 2.3 Given the conditional true statement: If you like donkers, then you will like torks. What conclusions can you make from the following statements? Mr. Mack likes torks. Mrs. Mack likes donkers. Mr. Mack does not likes donkers.

p  q q  r p  r UNIT 2 Deductive Reasoning 2.3 Law of Syllogism: Allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement. Law of Syllogism If p  q and q  r are true statements, then p  r is a true statement. p  q q  r p  r If a quadrilateral is a square, then it contains four right angles. If a quadrilateral contains four right angles, then it is a rectangle.

UNIT 2 Deductive Reasoning 2.3 The law of syllogism allows you to effectively “combine” two conditional statements into one conditional statement. Combine each pair of conditional statements into one conditional statement. If you liked the movie then you saw a good movie If you saw a good movie then you enjoyed yourself. If two lines are not parallel then they intersect. If two lines intersect then they intersect at a single point. If you like the ocean then you will like Florida. If you vacation at the beach then you like the ocean.

UNIT 2 Deductive Reasoning 2.3 What conclusion(s) can be made from each set of statements If a person visits the zoo they will see animals. Karla goes to the zoo. Richard sees animals If it snows then we will miss school. If we miss school then practice is cancelled. Practice is canceled. If you drive over the speed limit, then you will receive a ticket Sarah received a ticket.

Conditional statement p→q Converse statement q→p UNIT 2 Biconditionals and Definitions 2.4 The converse of a conditional statement is when the hypothesis and conclusion are exchanged. Conditional statement p→q Converse statement q→p If you study more then your grades improve The converse is :

UNIT 2 Biconditionals and Definitions 2.4 Write the converse for each conditional statement. If you go to the zoo, then you will see animals. If you drive slower, then your gas mileage will improve. If it is a triangle, then it is a 3-sided polygon.

UNIT 2 Biconditionals and Definitions 2.4 A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “p → q” and “if q → p.” The biconditional “p if and only if q” can also be written as “p iff q” or p  q. Write the statements as biconditionals. If you go to the zoo, then you will see animals.

UNIT 2 Biconditionals and Definitions 2.4 If you drive slower, then your gas mileage will improve. If it is a triangle, then it is a 3-sided polygon.

UNIT 2 Biconditionals and Definitions 2.4 The conditional and converse each have a truth value. The truth value of the conditional and converse may or may not be the same. Determine the truth value of each conditional and converse If you drive slower, then your gas mileage will improve. If the point is a midpoint, then it divides a segment into two congruent segments.

UNIT 2 Biconditionals and Definitions 2.4 If both the conditional and converse are both TRUE, Then the biconditional is also TRUE Hint…to determine if a biconditional is true, see if you can find a counterexample to the conditional or converse statement. Determine if the biconditonals are TRUE or FALSE. A figure is a triangle iff it is a 3-sided polygon. A figure is square iff it has 4 right angles.

UNIT 2 Biconditionals and Definitions 2.4 In geometry, biconditional statements are used to write definitions. Write each definition as a biconditional. A pentagon is a five-sided polygon. A figure is a pentagon if and only if it is a 5-sided polygon. A right angle measures 90°. An angle is a right angle if and only if it measures 90°.

An apple is a fruit that contains seeds. UNIT 2 Biconditionals and Definitions 2.4 To determine if a definition is good, evaluate the biconditional. An apple is a fruit that contains seeds.

HOMEWORK: 2.3(91): Worksheets 2.4(99): 11,15,16,19,21,23,29,37,39,43