 When dealing with compound statements, it's helpful to produce a truth table, and the truth table consists of the following. Every possible truth value of the simple statements is tabulated, and once we have these things organized, the truth value of the compound statement is computed. So we might begin by producing our truth tables for the basic compound statements. For negations, we have a statement x and its negation, not x. The statement itself can either be true or false, and the negation will have the opposite truth value. So if the statement is true, not x will be false, and if the statement is false, the negation will be true. If we have a conjunction, we have two statements, x and y, and our conjunction. Now to make a complete truth table, we have to include every possible assignment of the truth values for x and y. So x might be true, and y could either be true or false, and in the second case, don't forget, x is still true. x might also be false, and y could be either true or false. And again, remember in our second case, x is still false, and remember our conjunction is true only when both x and y are true. So of all these four cases, the only time the conjunction is true is the first case, and it's false in all the others. How about the disjunction? Again, we have two statements. An x could be true and y could be true, or x could be true and y false. And similarly, x could be false and y true or false. And a disjunction only needs one of its components to be true. So the disjunction will be true in all these cases, and the only time it will be false is when both statements are false. The conditional is a little strange, so let's take a look at that. So again, x and y could either be true or false, so let's list every possible combination. And remember that the conditional is false only when the antecedent is true and the consequent is false. So this case is the only time when the conditional is false, and it's true in every other case. How about something more complicated, like a and b implies b? So we want to produce a truth table for the compound statement, so we'll make it the final column of our table. Now there are two simple statements in this compound statement. There's a and there's b, so we'll make those the first two columns. Since a could be true or false and b could be true or false, we'll list all possible combinations. And since a conjunction b is a component of our compound statement, we'll include a column for the conjunction. And it's helpful to think about computing a truth value, and to do this most easily, we'll make each column correspond to one computation. So we can compute the truth value of the conjunction a and b. And since this is a conjunction, the only time it's true is when both a and b are true, and that's this first case, and it's false in all the others. How about our conditional? So remember, the only time a conditional has a possibility of being false is when the antecedent is true. And in these three cases, the antecedent is going to be false, and so the conditional is automatically true. In this case, the antecedent is true, so we have to look at whether the consequent is true as well. And since the consequent is b and b is true, then we know that the conditional is also true. Or consider a compound statement like this. There are three simple statements, a, b, c, and so we'll make each of these a column. We're trying to find the truth value of a and b or c, so we'll make that our final column. There's this conjunction a and b, and so let's compute that in the middle. And we want to include every possible combination of truth and false for a, b, and c. We can compute the truth of the conjunction a and b, and the disjunction is going to be true if either c or a and b is true, and so we can compute that truth value. Or let's consider a truth table for this compound statement. There are two simple statements, a and b. We have the statement that we want to compute the truth value of. And again, if we think about each column of our truth table as corresponding to one computation, we should have columns for not a, not b, a and b, not a or not b. So we'll list every possible combination of truth values of our simple statements. The truth value of not a is the opposite of the truth value of a. The truth value of not b is the opposite of the truth value of b. The conjunction a and b depends on whether a and b are both true. The disjunction not a or not b only requires one of these to be true, and the conjunction a and b and not a or not b requires both of our components to be true, which never happens. And so it's always false. It's helpful to introduce two ideas. We say that the compound statement is a tautology if it is always true regardless of the truth values of its components. For example, a and b implies b is a tautology because when we constructed its truth table, the truth values of a and b made no difference. The statement was always true. On the other hand, a compound statement is a paradox if it is never true regardless of the truth values of its components. And so when we found the truth value of this statement, we found that regardless of the truth values of a and b, the compound statement was always false. And so that means this statement is a paradox.