Geometry Honors Section 2. 2 Geometry Honors Section 2.2 Introduction to Logic and Introduction to Deductive Reasoning

The figures at the right are Venn diagrams The figures at the right are Venn diagrams. Venn diagrams are also called __________ diagrams, after the Swiss mathematician _______________________. Which of the two diagrams correctly represents the statement “If an animal is a whale, then it is a mammal”. Euler Leonard Euler   whale mammal

If-then statements like statement (1) are called If-then statements like statement (1) are called *___________ In a conditional statement, the phrase following the word “if” is the *_________. The phrase following the word “then” is the *_________. conditionals. hypothesis conclusion

If you interchange the hypothesis and the conclusion of a conditional, you get the *converse of the original conditional.

If an animal is a reptile than it is a snake. Example 1: Write a conditional statement with the hypothesis “an animal is a reptile” and the conclusion “the animal is a snake”. Is the statement true or false? If false, provide a counterexample. If an animal is a reptile than it is a snake. False

If an animal is a snake, then it is a reptile. Write the converse of the conditional statement. Is the statement true or false? If false, provide a counterexample. If an animal is a snake, then it is a reptile. True

Example 2: Consider the conditional statement “If two lines are perpendicular, then they intersect to form a right angle”. Is the statement true or false? If false, provide a counterexample.

Write the converse of the conditional statement Write the converse of the conditional statement. Is the statement true or false? If false, provide a counterexample.

When an if-then statement and its converse are both true, we can combine the two statements into a single statement using the phrase “if and only if” which is often abbreviated iff.   Example: Combine the two statements in example 2, into a single statement using iff.

Reasoning based on observing patterns, as we did in the first section of Unit I, is called inductive reasoning. A serious drawback with this type of reasoning is your conclusion is not always true.

*Deductive reasoning is reasoning based on Deductive reasoning logically correct conclusions always give a correct conclusion.

We will reason deductively by doing two column proofs We will reason deductively by doing two column proofs. In the left hand column, we will have statements which lead from the given information to the conclusion which we are proving. In the right hand column, we give a reason why each statement is true. Since we list the given information first, our first reason will always be ______. Any other reason must be a _________, _________ or ________. given definition postulate theorem

A theorem is a statement which can be proven A theorem is a statement which can be proven. We will prove our first theorems shortly.

Our first proofs will be algebraic proofs Our first proofs will be algebraic proofs. Thus, we need to review some algebraic properties. These properties, like postulates are accepted as true without proof.

Reflexive Property of Equality: Symmetric Property of Equality: a = a If a = b, then b = a

If a = b, then a+c = b+c If a = b, then a-c = b-c Addition Property of Equality: Subtraction Property of Equality: Multiplication Property of Equality: If a = b, then a+c = b+c If a = b, then a-c = b-c If a = b, then ac = bc

Division Property of Equality:

Substitution Property: If two quantities are equal, then one may be substituted for the other in any equation or inequality.

Distributive Property (of Multiplication over Addition): a(b+c) = ab + ac

Example: Complete this proof: Given Multiplication Property Distributive Property Addition Property Division Property

Example: Prove the statement: 1. Given