Geometry Honors Section 2.2 Introduction to Logic and Introduction to Deductive Reasoning
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Transcript of Geometry Honors Section 2.2 Introduction to Logic and Introduction to Deductive Reasoning
Geometry Honors Section 2.2
Introduction to Logic and Introduction to Deductive
Reasoning
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”.
whale
whale
mammal
mammal
Euler
Leonard Euler
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.
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.
crocodile :mplecounterexa
False
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.
TRUE
Write the converse of the conditional statement. Is the statement true or false?
If false, provide a counterexample.
larperpendicu are lines
then theangle,right a form tointersect lines twoIf
TRUE
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.
anglesright form tointersect
they ifflar perpendicu are lines Two
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. 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 ________.
givendefinition postulate theorem
A theorem is a statement which can be proven.
We will prove our first theorems shortly.
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
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:
cb c
a then 0,c and ba If
0?csay must weWhy
undefined! is 0by division Because
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:
45x6-2x .1 1. Given
3
10 )
x