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Proofs in Algebra Honors Sec 2.7 Proving Statements

Proofs in Algebra Honors Sec 2.7 Proving Statements. Definitions Axioms/Properties– statements about real numbers that are assumed to be true Hypothesis– states what is assumed to be true Conclusion– states something that logically follows an assumption. Statements a and b are real numbers

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Proofs in Algebra Honors Sec 2.7 Proving Statements

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  1. Proofs in AlgebraHonors Sec 2.7Proving Statements Definitions Axioms/Properties– statements about real numbers that are assumed to be true Hypothesis– states what is assumed to be true Conclusion– states something that logically follows an assumption

  2. Statements a and b are real numbers - b is a real number …and so forth Reasons Hypothesis Axiom of Additive Inverses … etc. How to Set up a Proof

  3. Axioms • Axiom of Closure • Commutative Axioms • Associative Axioms • Axioms of Equality • Reflexive Axiom • Symmetric Axiom • Transitive Axiom

  4. More Axioms • Distributive Axiom of Multiplication with respect to Addition • Cancellation Property of Opposites • Identity Axiom for Addition • Axiom of Additive Inverses • Property of the Opposite of a Sum You’ll also use Substitution!

  5. Statements 3m + (8 + 6m) = 3m + (6m + 8) = (3m + 6m) + 8 = (3 + 6)m + 8 = 9m + 8 5. 3m + (8+6m) = 9m + 8 Reasons Commutative for + 2. Associative for + 3. Distributive for x over + 4. Substitution 5. transitive Practice: Name the axiom that justifies each step

  6. Statements a and b are real numbers -b is real (a+b) + (-b) = a + [b + (-b)] = a + 0 = a 6. (a+b) + (-b) = a Reasons Hypothesis Axiom of Additive Inverses Associative Axiom of Additive Inverses Identity Axiom of Addition Transitive To Prove: For all real numbers a and b, (a + b) + (-b) = a

  7. Reasons Hypothesis Axiom of Additive Inverses Commutative Axiom for add. Associative Axiom of Additive Inverses Identity axiom for addition Transitive axiom of equality Statements a and b are real numbers - a is a real number (a + b) + (-a) = (b + a) + (-a) = b + [a + (-a)] = b + 0 = b (a + b) + (-a) = b To Prove: For all real numbers a and b, (a + b) + (-a) = b

  8. Reasons Hypothesis Axiom of additive inverses Identity axiom for addition Axiom of additive inverses Associative Axiom of additive inverses Identity axiom for addition Transitive axiom of equality Statements a is a real number - a is a real number -(-a) = -(-a) + 0 = -(-a) + (-a + a) = [-(-a) + (-a)] + a = 0 + a = a -(-a) = a To Prove: For all real numbers a, -(-a) = a

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