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Let's talk about cats. Cats can be either male or female. For convenience, we can use the letter M to refer to the class of male cats and F to refer to the class of female cats. Keep in mind that these two symbols do not represent numbers of cats. The number of male and female cats can change by the minute as new cats are born and old cats (regrettably) pass away. The letters stand for classes of cats—cats with specific characteristics. Instead of referring to male cats, we can just say "M."

We can also use other letters to represent the color of the cats: For example, T can refer to the class of tan cats, B can be the class of black cats, W the class of white cats, and O the class of cats of all "other" colors—all cats not in the class T, B, or W.

Finally (at least as far as this example goes), cats can be either neutered or unneutered. Let's use the letter N to refer to the class of neutered cats and U for the class of unneutered cats.

In conventional (numeric) algebra, the operators + and x are used to indicate addition and multiplication. In Boolean algebra, the same + and x symbols are used, and here's where things might get confusing. Everybody knows how to add and multiply numbers in conventional algebra, but how do we add and multiply classes?

Well, we don't actually add and multiply in Boolean algebra. Instead, the + and x symbols mean something else entirely.

The + symbol in Boolean algebra means a union of two classes. A union of two classes is everything in the first class combined with everything in the second class. For example, B + W represents the class of all cats that are either black or white.

The x symbol in Boolean algebra means an intersection of two classes. An intersection of two classes is everything that is in both the first class and the second class. For example, F x T represents the class of all cats that are both female and tan. As in conventional algebra, we can write F x T as F·T or simply FT (which is what Boole preferred). You can think of the two letters as two adjectives strung together: "female tan" cats.

To avoid confusion between conventional algebra and Boolean algebra, sometimes the symbols U and ∩ are used for union and intersection instead of + and x. But part of Boole's liberating influence on mathematics was to make the use of familiar operators more abstract, so I've decided to stick with his decision not to introduce new symbols into his algebra.

The commutative, associative, and distributive rules all hold for Boolean algebra. What's more, in Boolean algebra the + operator is distributive over the x operator. This isn't true of conventional algebra:

W + (B x F) = (W + B) x (W + F)

The union of white cats and black female cats is the same as the intersection of two unions: the union of white cats and black cats, and the union of white cats and female cats. This is somewhat difficult to grasp, but it works.

Two more symbols are necessary to complete Boolean algebra. These two symbols might look like numbers, but they're really not because they're sometimes treated a little differently than numbers. The symbol 1 in Boolean algebra means "the universe"—that is, everything we're talking about. In this example, the symbol 1 means "the class of all cats." Thus,

M + F = 1

This means that the union of male cats and female cats is the class of all cats. Similarly, the union of tan cats and black cats and white cats and other colored cats is also the class of all cats:

T + B + W + O = 1

And you achieve the class of all cats this way, too:

N + U = 1

The 1 symbol can be used with a minus sign to indicate the universe excluding something. For example,

1 – M

is the class of all cats except the male cats. The universe excluding all male cats is the same as the class of female cats:

1 – M = F

The other symbol that we need is the 0, and in Boolean algebra the 0 means an empty class—a class of nothing. The empty class results when we take an intersection of two mutually exclusive classes, for example, cats that are both male and female: