left arrowWhy do we need sets?
no left arrowWhat is a set?
no left arrowMembers of a set.
no left arrowSets of numbers
no left arrowSubsets

Why do we need sets?

In algebra, we understand the operations of addition (+) and multiplication (*). We can use these operations confident that the result will always be the same. For example, 1+2 always equals 3. We can depend on that fact whenever we do algebra. Also, according to the commutative law of addition, we can depend on the fact that 1+2 is always the same as 2+1.

For mathematicians, the question, "Why can we always depend on these results?" is very important. We can show graphically that 1+2 is the same as 2+1:

Diagram using dots to show that 1+2=2+1

We could also show graphically that 1,242,392 + 47,392 is the same as 47,392 + 1,242,392, but that would take a lot of time and paper. Also, we would have to show it for every combination of values to be sure that the commutative law always holds true for any whole numbers.

This shows the need for generalized proofs of the principles of algebra. In order to prove this result for all number, mathematicians use the concept of sets. If we can prove a principle for one set (all whole numbers, for example), then for any other set that has the same characteristics (real numbers, for example), the proof will still hold.