Logical Reasoning:ProofProve the theorem using the basic axioms of algebra.

1. If and then ac bc c a b 0, .ac bc c and 0 Given

1 1

cac

cbcF

HG b Mult. axiom of equality

1 1a b Inverse axiom of Mult.

a b Identity axiom of Mult.

Logical Reasoning:ProofProve the theorem using the basic axioms of algebra.

2. If then a c b c a b , .

a c b c Given

a c c b c c bg bg Add. axiom of equality

a c c b c c bgbg bgbgAssoc. axiom of Add.

a b 0 0 Inverse axiom of Add.

a b Identity axiom of Add.

Logical Reasoning:ProofFind a counterexample to show that the statement is not true.

3. a b a b2 2

1 2 1 22 2 1 4 3

5 3Because , you have shown one case in which the rule is false.

5 3

Logical Reasoning:ProofFind a counterexample to show that the statement is not true.

4. a b b a

1 2 2 1

1 1

Because , you have shown one case in which the rule is false.

1 1

Logical Reasoning:ProofFind a counterexample to show that the statement is not true.

5. a b b a

1 2 2 1

05 2.

Because , you have shown one case in which the rule is false.

05 2.

Logical Reasoning:ProofUse an indirect proof to prove that the conclusion is true.

6. If , then a b b c a c

a ca b c b

Assume the conclusion is false and contradict the given statement.

Assume the opposite of is true.a c

Add. axiom of equality

a b b c Comm. axiom of Add.This statement contradicts the original statement . Therefore, it is impossible that . Therefore is true.a b b c a c

a c

Logical Reasoning:Proof