Basic Techniques To Prove If/Then Statements

Let’s start by looking at the details of a process that goes on almost automatically in our minds hundreds of times every day–deciding whether something is true or false. Suppose you make the following statement:

When is your statement true? When is it false; that is, when could you be accused of lying?

The statement we are considering is a composite statement, and its two parts are the following simple statements:

As far as your trip is concerned, there are two possibilities:

Regarding the dinner, there are two possibilities as well:

Thus, we can consider four possibilities:

Case 1. You do go home and you do take your parents out to dinner. Your statement is true.

Case 2. You go home for the weekend, but you do not take your parents out to dinner. You have been caught lying! Your statement is false.

Cases 3 and 4. You cannot be accused of lying if you did not go home, but you did take your parents out to dinner, because they came to visit. If you did not go home, nobody can accuse you of lying if you did not take your parents out to dinner. It is very important to notice that you had not specified what you would do in case you were not going home (A is false). So, whether you did take your parents out to dinner or not, you did not lie.

In conclusion, there is only one case in which your statement is false–namely, when A is true and B is false. This is a general feature of statements of the form “If A, then B” or “A implies B.”

A statement of the form “If A, then B” is true if we can prove that it is impossible for A to be true and B to be false at the same time; that is, whenever A is true, B must be true as well.

The statement “If A, then B” can be reworded as “A is a sufficient condition for B” and as “B is a necessary condition for A.” The mathematical use of the words “sufficient” and “necessary” is very similar to their everyday use. If a given statement is true and it provides enough (sufficient) information to reach the conclusion, then the statement is called a sufficient condition. If a statement is an inevitable (certain) consequence of a given statement, it is called a necessary condition. A condition can be sufficient but not necessary or necessary but not sufficient.

As an example, consider the statement “If an animal is a cow, then it has four legs.” Having four legs is a necessary condition for an animal to be a cow, but it is not a sufficient condition for identifying a cow, as it is possible for an animal to have four legs and not be a cow. On the other hand, being a cow is a sufficient condition for knowing that the animal has four legs. Consult the James & James Mathematics Dictionary if you want to find out more about “sufficient” and “necessary” conditions.

All arguments having this form (called modus ponens) are valid. The expression “modus ponens” comes from the Latin ponere, meaning “to affirm.”

Very often one of the so-called truth tables* is used to remember the information just seen (T = true, F = false):

Because in a statement of the form “If A, then B” the hypothesis and the conclusion are clearly separated (part A, the hypothesis, contains all the information we are allowed to use; part B is the conclusion we want to reach, given the previous information), it is useful to try to write in this form any statement to be proved. The following steps can make the statement of a theorem simpler and therefore more manageable, without changing its meaning:

If you get stuck while constructing the proof, double-check whether you have overlooked some explicit or implicit information you are supposed to know and be able to use in the given context. As mentioned in the General Suggestions section, the proof of a statement depends on the context in which the statement is presented. The examples included in the next sections will illustrate how to use these suggestions, which at this point are somewhat vague, to construct some proofs.