Truth tables are a way of analyzing how the validity of statements (called propositions) behave when you use a logical or, or a logical and to combine them. Propositions are either completely true or completely false, so any truth table will want to show both of these possibilities for all the statements made. For all these examples, we will let p and q be propositions. They could be statements like I am 25 years old or it is currently warmer than 70°. Any statements that are either true or false. advertisement Negation not pNegation is the statement not p, denoted \(\neg p\), and so it would have the opposite truth value of p. If p is true, then \(\neg p\) if false. If p is false, then \(\neg p\) is true. Notice that the truth table shows all of these possibilities. Conjunction andConsider the statement p and q, denoted \(p \wedge q\). To analyze this, we first have to think of all the combinations of truth values for both statements and then decide how those combinations influence the and statement. In words:
The order of the rows doesnt matter as long as we are systematic in a way so that we do not miss any possible combinations of truth values for the two original statements p, q. Disjunction orYou may not realize it, but there are two types of ors. There is the inclusive or where we allow for the fact that both statements might be true, and there is the exclusive or, where we are strict that only one statement or the other is true. In math, the or that we work with is the inclusive or, denoted \(p \vee q\). When we want to work with the exclusive or, we are specific and use different notation (you can read about this here: the exclusive or). This shows in the first row of the truth table, which we will now analyze:
SummaryTo keep track of how these ideas work, you can remember the following:
Understanding these truth tables will allow us to later analyze complex compound compositions consisting of and, or, not, and perhaps even a conditional statement, so make sure you have these basics down! advertisement Continue reviewing discrete math topicsNext: Truth tables for the conditional and biconditional (implies, and iff) ![]() Subscribe to our Newsletter!We are always posting new free lessons and adding more study guides, calculator guides, and problem packs. Sign up to get occasional emails (once every couple or three weeks) letting you knowwhat's new! SUBSCRIBE Share this:
RelatedVideo |