Download Objective - To use properties of numbers in proofs.

Objective - To use properties of numbers in proofs.
Proof - An argument that proves a statement is
true either deductively or inductively.
Logical Reasoning
Deductive Reasoning
- process of demonstrating
that the validity of certain
statements can imply the
validity of statements that
follow.
All prime numbers greater
than 2 are odd.
37 is a prime number.
Therefore, 37 is odd.
Inductive Reasoning
- process of making
generalizations based on
observed data, patterns,
and past performance.
You have never seen a
pelican in the desert.
Therefore, pelicans
probably do not live in the
desert.
Conditionals (If-then Statements)
Deductive Reasoning
Inductive Reasoning
If your number is a prime
greater than 2, then it is odd.
If you have never seen
pelicans in the desert, then
they do not live there.
Hypothesis:
Hypothesis:
Your number is a prime
greater than 2.
Conclusion:
It is odd.
Deduction:
Certain!
Used in proofs!
You have never seen
pelicans in the desert.
Conclusion:
They do not live there.
Induction:
Likely!
Not often used in proofs!
Deductive Reasoning
Conjecture - a statement or conditional that one
is trying to prove.
Types of supportive statements used in proofs
1) Undefined terms - Terminology so fundamental
it defies definition.
ie: point, line, straight, etc.
2) Definitions - Statements defined by other terms.
ie: A quadrilateral is a 4 sided polygon.
3) Axioms (Postulates) - Property or statement
which is assumed to be true.
ie: Two points will determine a line.
4) Theorems - A property or statement which has
been proven to be true.
2
2
2
ie: Pythagorean Theorem a  b  c.
Closure Property
A set of numbers is said to be ‘closed’ if the
numbers produced under a given operation are
also elements of the set.
Tell whether the whole numbers are closed under
the given operation. If not, give a counterexample.
1) Addition
Closed
3) Multiplication
Closed
2) Subtraction Not Closed 4) Division Not Closed
5 - 7 = -2
2 8 = 0.25
Closure Property
A set of numbers is said to be ‘closed’ if the
numbers produced under a given operation are
also elements of the set.
Tell whether the integers are closed under the
given operation. If not, give a counterexample.
1) Addition
Closed
2) Subtraction Closed
3) Multiplication
Closed
4) Division Not Closed
2 8 = 0.25
Field Properties (Axioms) Used in Proofs
The Closure Properties
If a and b are rational, then a + b is rational.
If a and b are rational, then a b is rational.
The Commutative Properties
a+b=b+a , a b=b a
The Associative Properties
(a + b) + c = a + (b + c) , (a b) c = a (b c)
The Identity Properties
a+0=a , a 1=a
The Inverse Properties
1
a  (-a )  0 ,
a •  1 (where a  0)
a
The Distributive Property
a(b  c)  ab  ac
Additional Properties (Axioms) Used in Proofs
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 a c = b c.
Subtraction Property of Equality
If a = b, then a  c = b  c.
Other Properties
Reflexive Property
a=a
Symmetric Property
If a = b, then b = a.
Transitive Property
If a = b and b = c, then a = c.
Example of Direct Proof (Deductive)
Prove: If a = b, then -a = -b.
Statement
Reason
a=b
a + (-b) = b + (-b)
a + (-b) = 0
(-a) + [a + (-b)] = 0 + (-a)
[(-a) + a] + (-b) = 0 + (-a)
0 + (-b) = 0 + (-a)
-b = -a
-a = -b
Given
Addition Property of Equality
Inverse Property
Addition Property of Equality
Associative Prop. of Addition
Inverse Property
Identity Property of Addition
Symmetric Property