 A useful concept is that of logically equivalent. We say that two statements are logically equivalent if they have the same truth values. We can most easily verify logical equivalents using a truth table. For example, let's consider the conditional if a, then b. And we might look at it this way. Since we either have a, or we don't, we might interpret this as follows. If we have a, then we also have b. On the other hand, if we don't have a, well, who knows. The conditional we have doesn't tell us anything about what happens if we don't have a. So it seems we either have b or we don't have a. And so the question you might ask yourself is, is the statement if a, then b the same as saying either have b or we don't have a? To answer this question, let's compare the truth tables for if a, then b, and not a or b. In either case, there's only two simple statements, a and b, and they might be true or false. We want to compute the truth value of if a, then b. We also want to find the truth value of not a or b. And as an intermediate step, it may help to compute the truth value of not a. So we'll complete our truth table. And the thing to notice is that if a, then b, and not a or b have the same truth values at the same time. If one of them is true, so is the other. If one of them is false, so is the other. And since they have the same truth values, they are the equivalent statements. What this means is that we can rewrite any conditional as a disjunction. So let's rewrite as a disjunction. If a number is prime, then it is not divisible by two. So let's think about that. Our antecedent is a number is prime, while the consequent is is not divisible by two, or more grammatically, the number is not divisible by two. So we found that the conditional if a, then b is equivalent to the disjunction not a or b. And so we can say that either a number is not prime, or it is not divisible by two. Now as we'll see, it's not quite as useful to turn a conditional into a disjunction, but it is more useful to keep in mind that any conditional if a, then b gives rise to three related statements. So for example, the converse if b, then a, the inverse if not b, then not a, and the counter positive if not b, then not a. So let's try to compare the conditional, its converse, inverse, and counter positive. To do that, we'll construct the truth tables. So we have our conditional if a, then b, our converse if b, then a, our inverse if not a, then not b, and our counter positive if not b, then not a. Now there's only two simple statements here, a and b, so we can set up their possible values. We also want to find the negation of a and the negation of b, and first we'll find the truth value of the conditional if a, then b, and let's find the truth value of the converse, inverse, and counter positive. And if we focus on the four conditionals, we see right away that only the conditional and its counter positive have the same truth values. The converse is sometimes true when the conditional is false, and sometimes false when the conditional is true. And so this leads to the following very important theorem. The converse and inverse of a conditional are not logically equivalent, but the counter positive is logically equivalent. And as we'll see, this turns out to be a fantastically useful result.