Download Geometry Honors Section 2.2 Introduction to Logic

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 __________
Euler
diagrams, after the Swiss mathematician
_______________________.
Leonard Euler
Which of the two diagrams correctly represents
the statement
“If an animal is a whale, then it is a mammal”.
whale
mammal
whale
mammal
If-then statements like statement
(1) are called *___________
conditionals.
In a conditional statement, the
phrase following the word “if” is
hypothesis The phrase
the *_________.
following the word “then” is the
*_________.
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.
False
counterexample : crocodile
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.
If two lines intersect to form a right angle, then the
lines are perpendicular
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.
Two lines are perpendicular iff they
intersect to form right angles
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 logically correct conclusions
Deductive reasoning
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
given Any other reason must
always be ______.
theorem
be a _________,
definition _________
postulate or ________.
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:
a=a
Symmetric Property of Equality:
If a = b, then b = a
Addition Property of Equality:
If a = b, then a+c = b+c
Subtraction Property of Equality:
If a = b, then a-c = b-c
Multiplication Property of Equality:
If a = b, then ac = bc
Division Property of Equality:
If a  b and c  0, then a  b
c
c
Why must we say c  0?
Because division by 0 is undefined!
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. 2x - 6  5x  4 1. Given
 10
) x
3