Video: Truth Values of Conditional Statements

Let 𝐴 be the hypothesis β€œπ‘₯ + 3 = 3 + π‘₯” and 𝐡 be the conclusion β€œπ‘₯ is prime.” The conditional statement 𝐴 β‡’ 𝐡 reads, β€œif π‘₯ + 3 = 3 + π‘₯, then π‘₯ is prime.” Is this true or false? The converse statement 𝐡 β‡’ 𝐴 reads, β€œif π‘₯ is prime, then π‘₯ + 3 = 3 + π‘₯.” Is this true or false? The inverse statement ¬𝐴 β‡’ ¬𝐡 reads, β€œIf π‘₯ + 3 β‰  3 + π‘₯, then π‘₯ is not prime.” Is this true or false? The contrapositive statement ¬𝐡 β‡’ ¬𝐴 reads, β€œif π‘₯ is not prime, then π‘₯ + 3 β‰  3 + π‘₯.” Is this true or

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Video Transcript

Let 𝐴 be the hypothesis β€œπ‘₯ plus three equals three plus π‘₯” and 𝐡 be the conclusion β€œπ‘₯ is prime.” The conditional statement 𝐴 then 𝐡 reads, β€œif π‘₯ plus three equals three plus π‘₯, then π‘₯ is prime.” Is this true or false? The converse statement if 𝐡 then 𝐴 reads, β€œif π‘₯ is prime, then π‘₯ plus three equals three plus π‘₯.” Is this true or false? The inverse statement if not 𝐴, then not 𝐡 reads, β€œif π‘₯ plus three does not equal three plus π‘₯, then π‘₯ is not prime.” Is this true or false? The contrapositive statement if not 𝐡, then not 𝐴 reads, β€œif π‘₯ is not prime, then π‘₯ plus three is not equal to three plus π‘₯.” Is this true or false?

In this question, we have a series of if p-then q type statements. One helpful tool for solving these types of problems is using a truth table. We’ll consider the possibilities of our hypothesis being true and our hypothesis being false. In the two cases where our hypothesis is true, the conclusion can be true or false. In the two cases where our hypothesis is false, we’ll have a true and false conclusion. If the hypothesis is true and the conclusion is true, the conditional statement if p, then q is true. If the hypothesis is true but the conclusion is false, the statement is false.

The third line of the truth table is a little bit tricky because if the conclusion is true, even if the hypothesis is false, as a whole, the if-then statement is true because the conclusion is true. And finally, if both the conclusion and the hypothesis are false, the if-then statement is true. The only place in conditional statements where the conditional truth value is false is if you start with a true hypothesis and end up with a false conclusion. But let’s see how these play out in each of these four circumstances.

For the conditional statement, we have if π‘₯ plus three equals three plus π‘₯, and we know that that is true for any value of π‘₯, π‘₯ plus three equals three plus π‘₯. And now, we need to consider if this statement π‘₯ plus three equals three plus π‘₯ mean that π‘₯ is prime. To find out if this is true, we can try to think of a counterexample. The counterexample would be a place where π‘₯ plus three equals three plus π‘₯, but π‘₯ is not prime. For example, eight plus three is equal to three plus eight, but eight is not prime. And so, we have a case where the conditional is true, but the conclusion is false, which makes the if-then statement false.

Moving on to the converse statement if 𝐡, then 𝐴, our starting point is if π‘₯ is prime. We can choose any prime number. Let’s choose two. So that’s true. If π‘₯ is prime, we chose two. That’s a true statement. Then two plus three is equal to three plus two, which is also true. We want to know is there any place where we could plug in a prime number for π‘₯ such that π‘₯ plus three does not equal three plus π‘₯? No. This is because the statement π‘₯ plus three equals three plus π‘₯ is always true. Here’s a good time to think about that tricky truth value I talked about earlier. If p is false, but q is true.

For example, what if we plugged in the statement π‘₯ equals eight? Well, π‘₯ is not prime. So the first bit would be false. But eight plus three does equal three plus eight. And so, the conclusion would have to be true. Even if π‘₯ is not prime, so even if the statement π‘₯ is prime is false, the conclusion π‘₯ plus three equals three plus π‘₯ still stands. And this converse statement if 𝐡, then 𝐴 must be true. At this point, we could keep going with the same strategy. But there’s something we can remember about the relationship between conditional, contrapositive, converse, and inverse statements.

If the conditional statement is true, then the contrapositive is true. And likewise, if the conditional statement is false, then the contrapositive is false. We can also say that if the converse is true, then the inverse is true. Our third sentence is the inverse statement if not 𝐴, then not 𝐡. We know that the inverse must be true because the converse statement was true. And the converse statement and inverse statement have the same truth value. Let’s take a closer look at why.

We have the condition if π‘₯ plus three is not equal to π‘₯ plus three. This statement will always be false because π‘₯ plus three is equal to three plus π‘₯. If the condition is false, no matter what the conclusion is, if p-then q must be true. In other words, there’s nothing we could plug in for π‘₯ plus three not equal to three plus π‘₯ that would then be prime. If our condition is always false, it doesn’t matter if the conclusion is true or false. And finally, we’ll deal with the contrapositive statement, which says, if not 𝐡, then not 𝐴.

The contrapositive has the same truth value as the conditional statement. Our conditional statement was false, which means the contrapositive is also false. We can prove this is false by thinking of a counterexample, a place where π‘₯ is not prime. So we can use eight. Eight is not prime. So that makes the if π‘₯ is not prime true. The then statement would say then eight plus three is not equal to three plus eight. What we’ve just found is a true-then false. And we know if true-then false statements must be false.

To solve questions like these, it’s good to have multiple strategies from a truth table to memorising facts about the relationship between these four types of statements. In addition to that, it’s good to plug in values to help you see what’s happening. In this case, we have a false conditional, a true converse, a true inverse, and a false contrapositive.

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