Applied Discrete Structures

In this section, we will list the most basic equivalences and implications of logic. Most of the equivalences listed in Table Table 3.4.3 should be obvious to the reader. Remember, 0 stands for contradiction, 1 for tautology. Many logical laws are similar to algebraic laws. For example, there is a logical law corresponding to the associative law of addition, \(a + (b + c) = (a + b) + c\text{.}\) In fact, associativity of both conjunction and disjunction are among the laws of logic. Notice that with one exception, the laws are paired in such a way that exchanging the symbols \(\land\text{,}\) \(\lor\text{,}\) 1 and 0 for \(\lor\text{,}\) \(\land\text{,}\) 0, and 1, respectively, in any law gives you a second law. For example, \(p \lor 0\Leftrightarrow p\) results in \(p \land 1 \Leftrightarrow p\text{.}\) This is called a duality principle. For now, think of it as a way of remembering two laws for the price of one. We will leave it to the reader to verify a few of these laws with truth tables. However, the reader should be careful in applying duality to the conditional operator and implication since the dual involves taking the converse. For example, the dual of \(p \land q\Rightarrow p\) is \(p \lor q \Leftarrow p\text{,}\) which is usually written \(p\Rightarrow p \lor q\text{.}\)

Some of the logical laws in Table Table 3.4.4 might be less obvious to you. For any that you are not comfortable with, substitute actual propositions for the logical variables. For example, if \(p\) is “John owns a pet store” and \(q\) is “John likes pets,” the detachment law should make sense.