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.
Negation not p
Negation 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.
Consider 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.
You 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:
To 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!
Continue reviewing discrete math topics
Next: Truth tables for the conditional and biconditional (implies, and iff)
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