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One of the great things about binary is that it’s very efficient at handling special operations: namely, logical operations. Four basic logical operations exist although additional operations are derived from the basic four operations. Three of the operations — AND, OR, and XOR — compare two binary digits (bits). The fourth (NOT) works on just a single bit.

The following list summarizes the basic logical operations:

 AND: Compares two binary values. If both values are 1, the result of the AND operation is 1. If one or both of the values are 0, the result is 0.

 OR: Compares two binary values. If at least one value is 1, the result of the OR operation is 1. If both values are 0, the result is 0.

 XOR: Compares two binary values. If one of them is 1, the result is 1. If both values are 0 or if both values are 1, the result is 0.

 NOT: Doesn't compare two values but simply changes the value of a single binary value. If the original value is 1, NOT returns 0. If the original value is 0, NOT returns 1.

Table 3-2 summarizes how AND, OR, and XOR work.

TABLE 3-2 Logical Operations for Binary Values

First Value	Second Value	AND	OR	XOR
0	0	0	0	0
0	1	0	1	1
1	0	0	1	1
1	1	1	1	0

Logical operations are applied to binary numbers that have more than one binary digit by applying the operation one bit at a time. The easiest way to do this manually is to line the two binary numbers on top of one another and then write the result of the operation beneath each binary digit. The following example shows how you would calculate 10010100 AND 11011101:

10010100AND 11011101 10010100

As you can see, the result is 10010100.