**Set theory** constitutes most of the foundation of modern mathematics, and was formalized in the late 1800s. Set theory describes some very fundamental and intuitive ideas about how things called "elements" or "members" fit together into groups. Despite the apparent simplicity of the ideas, set theory is quite rigorous. In seeking to eliminate all arbitrariness in their theories, mathematicians have fine-tuned set theory to an impressive degree over the years.

In set theory a *set* is any well-defined group of elements or members. Sets are usually symbolized by italicized capital letters like *A* or *B*. If two sets contain the same members, they can be shown as equivalent with an equal sign.

The contents of a set can be described in simple English: *A* = all terrestrial mammals. Contents can also be listed within brackets: *A* = {bears, cows, pigs, etc.} For large sets, ellipsis may be employed, where the pattern of the set is obvious. For example, *A* = {2, 4, 6, 8... 1000}. One type of set has zero members, the set known as the *empty set*. It is symbolized by a zero with a diagonal line ascending left to right. Though seemingly trivial, it turns out to be quite important mathematically.

Some sets contain other sets, therefore being labeled *supersets*. The contained sets are *subsets*. In set theory, this relationship is referred to as "inclusion" or "containment," symbolized by a notation that looks like the letter **U** rotated 90 degrees to the right. Graphically, this can be represented as a circle contained within another, larger circle.

Some common sets in set theory include N, the set of all natural numbers; Z, the set of all integers; Q, the set of all rational numbers; R, the set of all real numbers; and C, the set of all complex numbers.

When two sets overlap but neither is completely embedded within the other, the whole thing is called a *union of sets*. This is represented by a symbol similar to the letter U, but slightly wider. In set notation, *A* U *B* means "the set of elements which are members of either *A* or *B*". Turn this symbol upside down, and you get the intersection of *A* and *B*, which refers to all elements which are members of both sets. In set theory sets can also be "subtracted" from each other, resulting in complements. For example, *B* - *A* is equivalent to the set of elements that are members of B but not A.

From the above foundations, most of mathematics is derived. Nearly all mathematical systems contain properties that can be described fundamentally in terms of set theory.