Set Theory and Various Types of Sets

Set theory is one of the important theories in mathematics. It can help in solving of various mathematical problems. The problems can be represented in the graphical form with the help of set theory. A set is nothing but a collection of objects. There can be intersection of sets and union of sets. On performing the intersection operation the common elements of both the sets are got. The union of two sets gives all the elements present in both the sets. Basically a set can consist of various objects in it. But usually in mathematics sets usually deal with the objects which are related to mathematics.

The concept of set theory is very ancient. In the 1870’s itself good research was done on this topic and considerable progress was made. A Venn diagram is best used to represent the operations on sets. The process of intersection and union can be easily represented on the Venn diagram. There can be bigger set and a smaller which is part of it is called its subset. This can be explained with the help of an example. If a set contains the elements {1, 2, 3, 4} and there is another set which contains the elements {1, 2} then the latter set is subset of the former set. It simply means the all the elements present in the second set are also present in the first set. First set covers the whole of the second set in it.

The compliment of a set is nothing but a set containing elements which are present in the universal set but are not present in the given set. The set complement can be explained with the help of an example. If there are elements like {a, b, c, d, e} in the universal set and elements in the given set whose compliment is to be found out, are {a, c, e} then the complement set is given by { b, d}. So, the elements b, d is contained in the universal set but is not present in the set whose compliment has to be found. So, these elements are part of the required set. The compliment of a set X can be represented by the notation X’. This is a simple notation and there is another method of representing the same. Instead of the apostrophe symbol the letter ‘c’ can also be used to represent compliment.