Sect 1.4 Theorems and Proofs

A theorem consists of two parts:

Hypothesis: states what is assumed to be true

Conclusion: states something which logically follows from the assumptions

To give a direct proof, you start with its hypothesis and by a logical chain of steps arrive at its conclusion. **Be sure to look at all theorems and properties in the text.

Ex. If a + (-c) = b + (-c), then a = b.

*In an if-then statement, the ‘if’ part is the hypothesis.

Statement Reason

1. a + (-c) = b + (-c) 1. hypothesis

2. –c is a real number 2. axiom of additive inverse

3. a = b 3. cancellation property of addition

Ex. If x = a + (-b), then x + b = a

Statement Reason

1. x = a + (-b) 1. hypothesis

2. x + b = x + b 2. reflexive

3. x + b = [a + (-b)] + b 3. substitution

4. = a + [(-b) + b] 4. associative for addition

5. = a + 0 5. additive inverse

6. = a 6. identity for addition

7. x + b = a 7. transitive

*Proof writing of this nature is not easy. Be sure that you know the theorems, axioms, and properties. This will help you a great deal. Do not give up or get frustrated. It takes time to learn the techniques.