Objective - To use properties of numbers in proofs.

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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
Inductive Reasoning
- process of demonstrating that the validity of
certain statements can imply the validity of
statements that follow.
- process of making generalizations based on
observed data, patterns, and past performance.
You have never seen a pelican in the desert.
All prime numbers greater than 2 are odd.
Therefore, pelicans probably do not live in
the desert.
37 is a prime number.
Therefore, 37 is odd.
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Conditionals (If-then Statements)
Deductive Reasoning
Inductive Reasoning
If you have never seen pelicans in the desert,
then they do not live there.
If your number is a prime greater than 2, then
it is odd.
Hypothesis
Your number is a prime greater than 2.
You have never seen pelicans in the desert.
Conclusion
They do not live there.
It is odd.
Deduction
Certain!
Likely!
Not often used in proofs!
Used in proofs!
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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.
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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.
3) Multiplication
1) Addition
Closed
Closed
2) Subtraction
4) Division
Not Closed
Not Closed
5 - 7 -2
2 8 0.25
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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.
3) Multiplication
1) Addition
Closed
Closed
2) Subtraction
4) Division
Closed
Not Closed
2 8 0.25
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Field Properties (Axioms) Used in Proofs
The Closure Properties
If a and b are rational, then a b is rational.
The Commutative Properties
a b b a
The Associative Properties
(a b) c a (b c)
The Identity Properties
a 0 a
The Inverse Properties
The Distributive Property
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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
Subtraction Property of Equality
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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.
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Example of Direct Proof (Deductive)
Prove If a b, then -a -b.
Statement
Reason
a b
Given
a (-b) b (-b)
Addition Property of Equality
a (-b) 0
Inverse Property
(-a) a (-b) 0 (-a)
Addition Property of Equality
(-a) a (-b) 0 (-a)
Associative Prop. of Addition
Inverse Property
0 (-b) 0 (-a)
-b -a
Identity Property of Addition
Symmetric Property
-a -b