The idea behind set theory is to help students, and prospecting mathematicians to establish logic behind certain concepts of basic mathematics. With this foundation, comes the basis of solving future and more advanced exercises such as topology, complex analysis and real analysis. Therefore, with a deeper understanding of basic mathematics that comprises set theory, the future of a mathematician is well on course.
A set is a well defined collection of objects referred to as members or elements of a set. A set is denoted by upper case letters, while its members are denoted by lower case letters (Jech, 1978).
For instance, let be a set and let denote the object. When p is the element or member of A, it is denoted by and this is read as. If both p and q belong to A, it is expressed as. If p is not a member of A, it is expressed as, , read as is not a member of or does not belong to.
Members of a set are normally enclosed in curly parenthesis given as. A set is, therefore, specified by either listing all its members or by stating the properties characterizing each of its members.
Sets can contain the following elements:
1. Singleton sets: set containing only one member
2. Set of natural numbers:
3. Set of even natural numbers:
4. Set of integers:
A set may also be divided into finite or infinite, depending on whether it contains a finite or infinite number of members. Therefore, if A is a finite set, the number of members in A is denoted by
Let S be a specified set. Any set A each of whose elements is also an element of S is said to be contained in S and is called a subset of S. For example, if , then the above are subsets of
Venn Diagrams and Set Operations
Let A and B be given sets. Then, the set of all elements belonging to both A and B is called the intersection of A and B. This is denoted by, read as intersection B. Therefore,
Let A and B be any two sets. Then, the union of A and B consist of all those members in either A or B. Union of A and B is denoted by, read as A union B. Therefore,
Other binary operations on sets involve the difference, which subtracts all elements which are B from A. Such that,
Sets are sometimes represented in the form of Venn diagrams so as to represent sets visually and their operations (Froman & Pyk, 1972).
There are many types of sets. Namely:
a. Generalized set theory which allows for the modification of atoms
b. Hyper set theory which fully explains the axiomatic set theory (Suppes, 1960)
c. Constructive set theory which helps in easily solving in the field of mathematics
d. Set theory is also important in the understanding of basic computer programming such as SQL
As observed from above, it is clear that set theory is important in the day to day professions around the world. Without scientists and software engineers, the world would be still held up in the old ages before civilization. This area of study, as much as it may appear abstract, should be given a lot of emphasis.
Froman, R., & Pyk, J. (1972). Venn diagrams. New York: Crowell.
Jech, T. J. (1978). Set theory. New York: Academic Press.
Suppes, P. (1960). Axiomatic set theory. Princeton, N.J.: Van Nostrand.