Logical Expressions

Conjunction: AND

The conjunction P AND Q (in formal logic often written as PλQ) evaluates to true if both P and Q evaluate to true. The application of this type of expression is straightforward, as we can use it to say that a specific action X must be taken in case both P and Q are true. For instance,

Disjunction: OR

The disjunction P OR Q (in formal logic often written as PηQ) evaluates to true if either P or Q is true. Disjunctions are easier to satisfy than conjunctions; in Table 1, PηQ evaluates to true in three of the four combinations of P's and Q's truth values, whereas the conjunction PλQ is true only in one of the four combinations.

Exclusive or: XOR

The exclusive or expression P XOR Q (sometimes written as P⊕Q) evaluates to true if one of the operands P or Q is true and the other is false. The XOR is not elementary like OR and AND, as it can be rewritten as

• (P AND NOT(Q)) OR (NOT(P) AND Q)

XOR expressions are used less frequently than conjunctions and disjunctions. They have some interesting applications, however. One of these is as a means to quickly convert between various series of operator values. To illustrate this, let the series 1101 represent four logical operands, P, Q, R and S, with a 1 representing true and a 0 representing false. XORing this series with another series, say 0101, yields 1101 XOR 0101 = 1000. Interestingly, if we now XOR, the result (1000) again with the series 0101, we obtain the original series back: 1000 XOR 0101 = 1101. Since series of 1s and 0s are used to represent numbers in binary form and since computers can manipulate binary numbers very rapidly, XORing is often used in computer programs that require certain properties to flip back and forth between two predefined values, using a third value as the XOR intermediary.

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