Understanding If-Then Statements Part 1

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.

if then

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 then q

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.

if p then q statements

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.

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