Question

If a = and b = then,

Answer 1

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Answer 2

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Three Ways to Prove “If A, then B.”

A statement of the form “If A, then B” asserts that if A is true, then B must be true also. If the statement

“If A, then B” is true, you can regard it as a promise that whenever the A is true, then B is true also.

Most theorems can be stated in the form “If A, then B.” Even if they are not written in this form, they

can be put into this form. For example, the statements

“Every group with 4 elements is abelian.” and “A group is abelian if it has 4 elements.”

can both be restated as: “If a group G has 4 elements, then G is abelian.”

There are three ways to prove a statement of form “If A, then B.” They are called direct proof, contrapositive proof and proof by contradiction.

DIRECT PROOF. To prove that the statement “If A, then B” is true by means of direct proof, begin

by assuming A is true and use this information to deduce that B is true. Here is a template. What comes

between the first and last line of course depends on what A and B are.

Theorem: If A then B.

Proof. Suppose A is true.

.

.

.

Therefore B is true.

CONTRAPOSITIVE PROOF. The idea is that if the statement “If A, then B” is really true, then

it’s impossible for A to be true while B is false. Thus, we can prove the statement “If A, then B” is true

by showing that if B is false, then A is false too. Here is a template.

Theorem: If A then B.

Proof. Suppose B is false.

.

.

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Therefore A is false.

PROOF BY CONTRADICTION. Again, if the statement “If A, then B” is really true, then it’s

impossible for A to be true while B is false. In other words, it is a contradiction to assume A is true and

B is false. Of course, since you have not proved “If A, then B” is a true statement, this contradiction is

not at all obvious. In the technique of proof by contradiction, you begin by assuming A is true and

B is false, and use this to deduce and obvious contradiction of from “C is true and C is false.” Here’s a

template.

Theorem: If A then B.

Proof. Suppose A is true and B is false.

.

.

.

Therefore C is true and C is false.

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