Objective - To use properties of numbers in proofs.

1. 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.

2. Hypothesis: Conclusion: Induction: 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: You have never seen pelicans in the desert. Your number is a prime greater than 2. Conclusion: They do not live there. It is odd. Deduction: Likely! Certain! Not often used in proofs! Used in proofs!

3. Deductive Reasoning ie: Pythagorean Theorem . 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.

4. 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 Not Closed 4) Division Not Closed 5 - 7 =-2 2 8 = 0.25

5. 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 Closed 4) Division Not Closed 2 8 = 0.25

6. If a and b are rational, then ab is rational. , ab=ba , (ab) c=a (b c) , a 1= a 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

7. If a = b, then a c = b c. If a = b, then a c = b c. 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

8. 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.

9. 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