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It's true, you know. There is such a thing as an algebra of sets. In fact, there's also an algebra of forms. There's even a Boolean algebra, which has to do with logic. So let's see what the algebra of sets is all about.

The algebra of sets is like ordinary algebra, but different. In arithmetic you add and subtract numbers and multiply and divide them. And in the algebra you perform analogous operations on sets. As a matter of fact, that's pretty much what any algebra is all about. It's a systematic way of dealing with objects of one kind or another. It means following certain rules. An algebra of sets, in particular, is a systematic way of working with sets. 

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In A Class Act with Sets we look informally at the nature of sets, the way they're defined, and the operations that can be performed on them. And in Getting a Number on Algebra we look at ordinary algebra, or the algebra of numbers. You might like to review these pages to freshen your memory.

The algebra of numbers is really a more systematic way of talking about operations and relations having to do with numbers.

In a similar way, the algebra of sets is a systematic way of handling problems relating to the operations on sets. It deals with procedures for carrying out "calculations" with sets involving their union, intersection, and complementation, and the relation inclusion. It deals with a development of the basic properties of these operations and their interrelations.

To be more technical, the algebra of sets is the set-theoretic analog of the algebra of numbers, which is concerned with addition, subtraction, multiplication, and division, and the relation less than.

The elements of the algebra of sets are certain equations that are always true in the algebra, regardless of which universal set and its subsets are represented. These equations are known as identities, or tautologies. A systematic approach develops theorems.

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Here we look at examples of proofs. But first I should explain what's meant by a proof and say what we're trying to prove. You might review Ordinary Logic for background.

To begin with, it's important to know we're dealing with statements about sets, and we're trying to prove they're true. The statements are called identities, or logical statements.

To see the difference between factual and logical sentences, look at the difference between the English sentence:

John is six feet tall.

and the sentence:

If John is six feet tall, and if being six feet tall means John is a basketball player, then John is a basketball player.

You can see that you wouldn't use logic to determine if John is six feet tall. This is a factual (or empirical) matter and logic won't help you a bit. You'd be better off using a tape measure.

As for the second sentence, no physical measurement will validate it. This is strictly a logical statement, pure and simple, and empirical observation won't help. In fact, you can change the parts and still have the same general sentence. For example, you can express similar things about Mary:

If Mary is five feet ten, and if being five feet ten says Mary is a tennis player, then Mary is a tennis player.

You can even use symbols in place of the people and their activities, yet the sentence, or rather the form of the sentence, remains the same:

If P, and if P then Q, then Q.

This statement is really a logical argument, and can be written as two premises and a conclusion:

P

If P then Q

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Q

To prove it, we can use truth tables, as we do here.

The last step is to see that our sentence is the conditional formed by the green column and the turquoise column and is always true, and this validates it.

When dealing with sets, the statements we make are called set-theoretic statements and derive their meaning from their properties. But again we distinguish between factual and logical statements -- in this case the logical statements take the form of identities, or theorems. It is these statements that are to be proved in the algebra of sets, and we rely on the properties of sets to prove them.

A primary theorem of the algebra of sets lists the basic properties of set operators, known as union and intersection, as we see here. The theorem has five sub-divisions, each of which has two parts. We only prove one of the parts but this should be enough to show how proofs are typically run.

The part is the so-called associative law for union:

A È (B È C) = (A È B) È C

The nature of the proof is to show that the set on the left of the equality sign (call it D, for convenience) is the same as the set on the right side (call it E, for now). We can do this by showing that every member of D is also in E, and conversely that every member of E is in D.

Proof: Suppose x Î D (i.e., x is a member of D).

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