In this post, we are going to examine the structure and truth table of the conditional statements or if-then statements. If-then statements are used often in mathematical proofs as well as real-life conversations. But before that, let us understand what proposition means.

A **proposition** is a declarative sentence that is either True (T) or false (F) but not both. If-then statements are composed of two propositions.

Let us consider the following statement by a father to his son.

“If you get an *A* in Calculus, then I’ll buy you a laptop.”

We can split this statement into two propositions *p* and *q* as follows.

*p*: You get an A in Calculus.

*q*: I’ll buy you a laptop.

In the if-then statement above, we call *p* the hypothesis and *q* the conclusion. Since *p* can be True or False and *q* be also True or False, all the possible combinations are shown in the following table.

We examine now the truth value of *if p then q* or did the break his promise or not. If he did, then *if p then q* is False, otherwise, True.

Notice that if *p* is True and *q* is True, that is, the son got an A in Calculus, and he bought him a laptop, then the father did not break his promise. Therefore, if *p* then *q* is True.

Now, if *p* is False and *q* is False, that is, the son did not get an A in Calculus and the father did not buy him a laptop, then the father did not break his promise. Therefore, *if p then q* is True.

If *p* is True and *q* is False, that is the son got an A in Calculus, and the father did not buy him a laptop, then the father broke his promise. That is, *if p then q* is False.

Lastly, if *p* is False and *q* is True, that is if the son did not get an A in calculus and the father still bought him a laptop, then the father did not break his promise. Remember, the father only told the son that if he got an A in Calculus, he will buy him a laptop. He did not tell him what will he do if he didn’t get an A. Therefore if p then q is still true. The complete truth table of if p then q is shown below.

As we can see, if p then q is false the hypothesis of the statement is True and the conclusion is False. In the next post, we will talk about the related conditional statements: converse, inverse, and contrapositive.