### Video Transcript

Let π΄ be the hypothesis βπ₯ plus
three equals three plus π₯β and π΅ be the conclusion βπ₯ is prime.β The conditional statement π΄ then
π΅ reads, βif π₯ plus three equals three plus π₯, then π₯ is prime.β Is this true or false? The converse statement if π΅ then
π΄ reads, βif π₯ is prime, then π₯ plus three equals three plus π₯.β Is this true or false? The inverse statement if not π΄,
then not π΅ reads, βif π₯ plus three does not equal three plus π₯, then π₯ is not
prime.β Is this true or false? The contrapositive statement if not
π΅, then not π΄ reads, βif π₯ is not prime, then π₯ plus three is not equal to three
plus π₯.β 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. Weβll 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, weβll 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 letβs 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 π΅, then π΄, our starting point is if π₯ is prime. We can choose any prime number. Letβs choose two. So thatβs true. If π₯ is prime, we chose two. Thatβs 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. Hereβs 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 π΅,
then π΄ must be true. At this point, we could keep going
with the same strategy. But thereβs 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 π΄, then not π΅. 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. Letβs 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, thereβs 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 doesnβt matter if the conclusion is true or false. And finally, weβll deal with the
contrapositive statement, which says, if not π΅, then not π΄.

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 weβve just found is a
true-then false. And we know if true-then false
statements must be false.

To solve questions like these, itβs
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, itβs good to
plug in values to help you see whatβs happening. In this case, we have a false
conditional, a true converse, a true inverse, and a false contrapositive.