Set theory is the foundation of modern mathematics, describing how elements fit into groups. Sets are well-defined groups of elements, symbolized by letters. Sets can contain other sets, with the contained sets being subsets. Common sets include natural, integer, rational, real, and complex numbers. Set union and intersection refer to elements that are members of both sets. Sets can also be subtracted to obtain complements. Most of mathematics is derived from set theory.
Set theory forms the bulk of the foundation of modern mathematics and was formalized in the late 1800s. Set theory describes some very basic 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 an effort to eliminate any arbitrariness in their theories, mathematicians have refined 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 such as A or B. If two sets contain the same members, they can be shown as equivalent with an equals sign.
The contents of a set can be described in plain English: A = all land mammals. Contents may also be listed in parentheses: A = {bears, cows, pigs, etc.} For large sets, ellipses may be used, 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 an empty set. It is symbolized by a zero with a diagonal line going up from left to right. Although seemingly trivial, it turns out to be quite important from a mathematical point of view.
Some sets contain other sets, so they are labeled as 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 fully incorporated into the other, the whole thing is called a set union. This is represented by a symbol similar to the letter U, but slightly wider. In set notation, AUB means “the set of elements which are members of A or B”. Turn this symbol upside down and you get the intersection of A and B, which refers to all elements that are members of both sets. In set theory, sets can also be “subtracted” from each other, obtaining complements. For example, B – A is equivalent to the set of elements which are members of B but not of A.
From the above foundations, most of mathematics is derived. Almost all mathematical systems contain properties that can basically be described in terms of set theory.
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