1. Objective - To use properties of numbers in proofs. Logical Reasoning Deductive ReasoningInductive 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. All prime numbers greater than 2 are odd. 37 is a prime number. Therefore, 37 is odd. You have never seen a pelican in the desert. Therefore, pelicans probably do not live in the desert. Proof - An argument that proves a statement is true either deductively or inductively.

2. Conditionals (If-then Statements) If your number is a prime greater than 2, then it is odd. Hypothesis: Your number is a prime greater than 2. Conclusion: It is odd. Deductive ReasoningInductive Reasoning Deduction: Certain! Used in proofs! If you have never seen pelicans in the desert, then they do not live there. You have never seen pelicans in the desert. They do not live there. Hypothesis: Conclusion: Induction: Likely! Not often used in proofs!

3. 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. ie: Pythagorean Theorem.

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. 1) Addition 2) Subtraction 3) Multiplication 4) Division Closed Not Closed 5 - 7 = - 2 Closed Not Closed 2 8 = 0. 25

5. 1) Addition 2) Subtraction 3) Multiplication 4) Division Closed Not Closed 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. Closure Property 2 8 = 0. 25

6. 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 + aa + b = b + a The Associative Properties The Identity Properties If a and b are rational, then a b is rational., a b = b a (a + b) + c = a + (b + c), (a b) c = a (b c) a + 0 = a, a 1 = a The Inverse Properties The Distributive Property

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

8. Other Properties Reflexive Property a = aa = 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. StatementReason a = bGiven a + (-b) = b + (-b)Addition Property of Equality a + (-b) = 0Inverse Property (-a) + [a + (-b)] = 0 + (-a)Addition Property of Equality [(-a) + a] + (-b) = 0 + (-a)Associative Prop. of Addition 0 + (-b) = 0 + (-a) Inverse Property -b = -aIdentity Property of Addition -a = -b Symmetric Property