Principles of Mathematics in Operations Research

1.3 The art of making proofs 5

Table 1.1. Operation table for A => B

A

True

True

False

False

B

True

False

True

False

A=>B

True

False

True

True

Formally speaking, A=> B means

1. whenever A is true, B must also be true.


2. B follows from A.


3. B is a necessary consequence of A.


4. A is sufficient for B.


5. A only if B.
   There are related statements to our primal assertion A =>• B:
   B =>• A: converse
   A =>• B: inverse
   B => A: contrapositive
   where A is negation (complement) of A.

1.3 The art of making proofs

This section is based on guidelines of how to read and make proofs. Our

pattern here is once again A =>• B. We are going to start with the forward-

backward method. After discussing the special cases defined in A or B in terms

of quantifiers, we will see proof by Contradiction, in particular contraposition.

Finally, we will investigate uniqueness proofs and theorem of alternatives.

1.3.1 Forward-Backward method

If the statement A =4> B is proven by showing that B is true after assuming
A is true (A -t B), the method is called full forward technique. Conversely, if
we first assume that B is true and try to prove that A is true (A <- B), this
is the full backward method.

Proposition 1.3.1 If the right triangle XYZ with sides x, y and hypotenuse
of length z has an area of ^- (A), then the triangle XYZ is isosceles (B). See
Figure 1.2.