REPRESENTATION OF SET

Set acts as a fundamental part of present-day mathematics. So, a set can be said as a well-defined collection of objects. The concept of set is used to define the concept of functions and relations. The knowledge of sets is needed to study geometry, sequences, probability, etc. 

In a set objects, elements and members of a set are synonymous terms. We usually denote sets by capital letters A, B, C, X, Y, Z, etc. The element of a set is represented by small letters a, b, c, etc. There are two methods of representing the set- Roster or tabular form and set-builder form. 

REPRESENTATION OF SET

As discussed above, the set can be represented by two methods-

• Roster/ Tabular form

• Set–builder form 

• Descriptive form

In roster form, elements of a set are written and are being separated by commas and enclosed within braces { }. For example- A set of all even positive integers less than 10 is given in roster form as {2, 4, 6, 8}. In set-builder form, all the elements of a set have a single property that is not possessed by any element outside the set. For example- consider the given set {a, e, i, o, u }. All the elements have a common property i.e. each one of them is a vowel in English alphabets and no other letter possesses this property. 

It is important to remember that while writing the set in roster form an element is not generally repeated i.e. all the elements are taken as distinct. For example- consider the set of letters which form the word ‘ SCHOOL’ is {S, C, H, O, L} or {H, O, L, C, S}. Here, the order in which the elements are listed is not at all important or relevant. 

Descriptive form:

The well-defined descriptions of a member of a set are stated in Descriptive statement form and wrapped in curly brackets.

For instance, consider the group of even numbers less than 15. It can be represented as ‘even numbers less than 15’ in statement form.

EMPTY SET- 

The empty set can be defined as the set which does not contain any element. The empty set is given by the symbol φ or {  }. For example-

Let A= {x:x is an even prime number greater than 2}. Here, A is an empty set because 2 is the only prime number. 

FINITE AND INFINITE SETS-

A set is referred to as a finite set when the set is empty or consists of a definite number of elements and when otherwise, the set is called infinite. For example-

Suppose G is the set of points on a line. Here, G is infinite.

Suppose A be the set of the days of the week. Here, A is finite. 

All infinite sets could not be described in roster form. For example- The set of real numbers cannot be described in this form, because the elements of the set do not follow any particular pattern.

EQUAL SET-

Two sets P and Q are called equal if they have exactly the same elements and we can write P=Q. If it is another way round, the set is called unequal and it can be written as P≠Q.

A set does not change if one or more elements of the set are repeated. To explain it better, let us take an example- the sets A={1, 2, 3} and B={ 2, 2, 1, 3, 3} are equal. Each element of A is in B and vice versa. Hence, we do not repeat any element in describing a set. 

DESCRIPTION METHOD OF A REPRESENTATION OF SET-

We have already discussed the way of representation of sets in set-builder form and roster form. Taking the same example which we have discussed of vowels. In the set {a, e, i, o, u}, it can be denoted by V,   V= { x:x is a vowel in the English alphabet}.

We describe the elements of the set by using a symbol x ( can use any symbol like y, z, etc ) followed by a colon “: “. After the colon, the characteristic property of the element of the set is written, and then enclose the whole description within the braces. Colon stands for “ such that”. Braces stand for “ the set of all”.  If you want to read the above description, then it is read as “ the set of all x such that x is a vowel of the English alphabet”. 

DESCRIPTION OF A SUBSET-

A set A is defined as a subset of set B if every element of A is also an element of B. Power of a set A is defined as the collection of all subsets of a set A. It is denoted by P(A). In P(A), every element is a set. For example- Suppose A= {1, 2}., then 

P(A)={ φ, {1}, {2}, {1,2} }

Talking about the universal set, it is represented by U. To explain this, let us take an example- Let’s take a set of integers, the universal set could be the set of rational numbers or the set R of real numbers. 

CONCLUSION 

Set is fundamental to understanding the relationships and functions in mathematics. The set is defined as the well-defined collection of objects. The set can be represented in two forms – roster form and set builder form. We have already discussed both of them in detail. Two sets A and B are equal if they have the same elements. Talking about the subset, A set A is said to be a subset of set B if every element of A is also an element B.