Python Programming: An Introduction to Computer Science: 3rd Edition by John Zelle

Author:John Zelle [Zelle, John]
Language: eng
Format: epub
Publisher: UNKNOWN
Published: 2018-07-29T23:00:00+00:00

18.4.21 Bool ean Algebra

All decisions in computer programs boil down to appropriate Boolean expressions. The ability to formulate, manipulate, and reason with these expressions is an important skill for programmers and computer scientists. Boolean expressions obey certain algebraic laws similar to those that apply to numeric operations . These laws are called Boolean logic or Boolean algebra.

Let's look at a few examples. The following table shows some rules of algebra with their correlates in Boolean algebra:

algebra Boolean algebra

a and false = = false

a and true== a

a+O=a a or false== a

From these examples, you can see that and has similarities to multiplication, or has similarities to addition, and 0 and 1 correspond to false and true . Here are some other interesting properties of Boolean operations . Anything ored with true is just true.

( a or True ) == True

Both and and or distribute over each other.

( a or (b and c) ) -( (a or b) and (a or c) ) ( a and (b or c) ) -( (a and b) or (a and c) )

A double negative cancels out.

( not (not a) ) == a

The next two identities are known as DeMorgan's laws .

( not(a or b) ) == ( (not a) and (not b) ) ( not(a and b) ) == ( (not a) or (not b) )

Notice how the operator changes between and and or when the not is pushed into an expression. One nice property of Boolean algebra is that simple identities of this sort can be easily verified with a truth table. Since there are always a finite combination of possible values for the variables, we can systematically list all of the possibilites and compute the value of the expressions. For example, the following table demonstrates DeMorgan's first law:

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