CHAPTER 2: DEDUCTIVE REASONING

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Presentation transcript:

CHAPTER 2: DEDUCTIVE REASONING Section 2-1: If-Then Statements; Converses

IF-THEN STATEMENT If-then statements, which are also called conditional statements or conditionals, are statements that include a hypothesis followed by a conclusion.

EXAMPLES OF IF-THEN If it rains after school, then I will give you a ride home. If B is between A and C, then AB + BC = AC. If I don’t pay attention in this class, then I will fail. I will become a 49ers fan if the Raiders move back to L.A.

REPRESENTING AN IF-THEN To represent if-then statements, we let p represent the hypothesis and let q represent the conclusion. Using p = hypothesis and q = conclusion, the base form of an if-then statement can be shown by: If p, then q. p: hypothesis q: conclusion

CONVERSE Converse: Reversing something as in position or order. The converse of a conditional is formed by interchanging the hypothesis and the conclusion.

CONVERSE EXAMPLE Statement: If p, then q. Converse: If q, then p. In short, flip the order of the hypothesis and conclusion to find the converse of a statement.

PRACTICE If points lie in 1 plane, then they are coplanar. State the converse of each conditional and tell whether each converse is true or false: If points lie in 1 plane, then they are coplanar. If points are collinear, then they all lie in one line. If x = -2, then 4x = -8.

STATEMENT VS. CONVERSE Statements and their converses do not say the same thing. Some true statements have converses that are not true. Example Statement: If I live in Castro Valley, then I live in the Bay Area. Converse: If I live in the Bay Area, then I live in Castro Valley. (False)

COUNTEREXAMPLE An if-then statement is false if an example can be found for which the hypothesis is true and the conclusion is false. Such an example is called a counterexample. Take the converse from the previous slide for example. I could live in the Bay Area but live in another city such as San Leandro.

PRACTICE If ab < 0, then a < 0. Provide a counterexample to show that each statement is false. If ab < 0, then a < 0. If a four-sided figure has four right angles, then it has four congruent sides. If a line lies in a vertical plane, then the line is vertical.

IF-THEN/CONDITIONAL STATEMENTS Conditional Statements or If-Then statements are not always written with the words if and then. General Form Example If p, then q. If 4x = 12, then x = 3 p implies q. 4x = 12 implies x = 3 p only if q. 4x = 12 only if x = 3 q if p. x = 3 if 4x = 12

BICONDITIONAL If a conditional statement and its converse are both true, they can be combined into a single statement using the words “if and only if”. Statements that contain the words “if and only if” are called biconditionals.

BICONDITIONAL The basic form of a biconditional can be found by: p if and only if q.

B is between A and C if and only if PRACTICE Write the pair of conditionals as a biconditional: If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. B is between A and C if and only if AB + BC = AC

CLASSWORK/HOMEWORK Classwork Pg. 34, Classroom Exercises 2-16 even Pg. 35, Written Exercises 2-28 even