Section 2.3: Deductive Reasoning Geometry 4/27/2017

Goals Use symbolic notation to represent logical statements Form conclusions by applying the laws of logic to true statements Geometry 4/27/2017

Symbolic Notation` Conditional Statement Converse Inverse p → q Converse q → p Inverse ~p → ~q Contrapositive ~q → ~p Biconditional p  q Geometry 4/27/2017

Example 1: Using Symbolic Notation Let p be “the value of x is -4” and q be “the square of x is 16.” Write p → q in words If the value of x is -4, then the square of x is 16 Write q → p in words If the square of x is 16, then the value of x is -4 Decide whether the biconditional statement p  q is true the conditional statement in part (a) is true, but the converse in part (b) is false Geometry 4/27/2017

Example 2: Writing an Inverse and a Contrapositive Let p be “today is Monday” and q be “there is school.” a) Write the contrapositive of p → q. ~q → ~p, if there is no school, then today is not Monday b) Write the inverse of p → q ~p → ~q, If today is not Monday, then there is no school Geometry 4/27/2017

Using The Laws of Logic Inductive Reasoning Deductive Reasoning Looking at several specific situations to arrive at a conjecture Deductive Reasoning Uses a rule to make a conjecture Geometry 4/27/2017

Laws of Deductive Reasoning Logical Argument Using facts, definitions, and accepted properties in a logical order Laws of Deductive Reasoning Law of Detachment Law of Syllogism Geometry 4/27/2017

Law of Detachment If p → q is a true statement and p is true, then q is true Geometry 4/27/2017

Law of Syllogism If p → q and q → r are true conditionals, then p → r is also true Geometry 4/27/2017