Section 2-1 Using Deductive Reasoning. If/then statements Called conditional statements or simply conditionals. Have a hypothesis (p) and a conclusion.

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Presentation transcript:

Section 2-1 Using Deductive Reasoning

If/then statements Called conditional statements or simply conditionals. Have a hypothesis (p) and a conclusion (q), "If p, then q:” p q

Example #1 If it is a Friday, then you will not have homework. Hypothesis (p): it is a Friday Conclusion (q): you will not have homework

Do not include the words “If” and “then” when naming the hypothesis and conclusion!

Sometimes a conditional statement is written without using "if" and "then".

Example #2 Perpendicular lines intersect at right angles. Can rewrite: If two lines are perpendicular, then the lines intersect at right angles.

Hypothesis (p): Conclusion (q): two lines are perpendicular the lines intersect at right angles If two lines are perpendicular, then the lines intersect at right angles.

converse formed by interchanging the hypothesis and conclusion of the conditional. Conditional : p q Converse : q p

Example #3 If two lines are perpendicular, then the lines intersect at right angles. Write the converse. If two lines intersect at right angles, then the lines are perpendicular.

A statement and its converse say different things. Some true conditionals may have a false converse.

counterexample an example that disproves a statement –O–Only need one counterexample to disprove a statement

Example #4 If Angela lives in Philadelphia, then she lives in Pennsylvania. True

Converse: If Angela lives in Pennsylvania, then she lives in Philadelphia. False Counterexample: She could live in Newtown and still live in PA.

biconditional A statement that contains the words “if and only if” p iff q Iff stands for “if and only if”  Used when a conditional and its converse are both true

Example #5 Congruent segments are segments that have equal lengths. Biconditional: Segments are congruent if and only if their lengths are equal.

inverse the negation of both the hypothesis and the conclusion of the conditional. –T–The denial of a statement is called a negation.

Conditional: p q Inverse: ~ p ~ q Read as not p then not q

Example #6 If two lines are perpendicular, then the lines intersect at right angles. If two lines are not perpendicular, then the lines do not intersect at right angles. Inverse:

contrapositive negation of both the hypothesis and conclusion of the converse Converse: q p Contrapositive: ~q ~p Read as not q then not p

Example #7 If two lines are perpendicular, then the lines intersect at right angles. Write the converse. If two lines intersect at right angles, then the lines are perpendicular.

Then write the contrapositive. If two lines do not intersect at right angles, then the lines are not perpendicular.