Sets Representation and Types

Sets Representation and Types:

What is a Set?

A set is a collection of well-defined and well-distinguished objects of our perception. The term “well-defined” implies a rule which helps one to identify whether a particular object belongs to the set or not. For example, the collection of “popular people” is not well defined because the same people may be popular to one but may not be so to some other person. Similarly, a collection of “good books on Science” is also not well defined. Such a collection cannot be a set.

Sets are usually denoted by capital letters of the alphabet and the members of the set are represented by lower case (small) letters. If a is a member of the set A, we write a ∈ A. If b is not a member of the set A, we write b ∉ A (a ∈ A is read as “a belongs to A” and b ∉ A is read as “b does not belong to A”). If a and b both belong to A, then we write a, b ∈ A.

Some Standard Sets:

Representation of a Set:

A set can be represented by the following methods-

(1) Tabular or Roster Form- In this method, the set is represented by listing all its members, separating them by commas and enclosing them in braces (curly brackets). For example-

(2) Rule Method or Set Builder Form- In this method, the members are not listed but the set is represented by specifying the rule or property following which the elements may be obtained uniquely. Thus, if A is the set consisting of the elements x satisfying the property p, then we write A = {x : x satisfies the property p} and is read as “A is the set of all elements x, where x is such that x satisfies the property p”.

Similarly, B = {x : x = n/(n² + 1), n ∈ N and n ≤ 4} implies B is the set of all elements x, where x is such that x = n/(n² + 1), n is a natural number and n is less than or equal to 4. When written in Roster form, the set would be-

The set of all real numbers lying between -2 and 2 (both inclusive) can be described in the Rule method as, {x : x ∈ R, -2 ≤ x ≤ 2}.

Types of Sets:

(1) Null Set- A set is said to be a null set or an empty set or a void set if it doesn’t have any element. A null set is denoted by the Danish letter Ø (oe). It is not to be confused with the Greek letter Φ (phi). In Roster form, Ø is also denoted by { }.

(2) Finite Set- A set consisting of a finite number of elements is said to be a finite set. Examples of a finite set are-

• Set of students of a class.
• Set of even numbers less than 50.
• {x : x = n/(n + 1), n ∈ N and n ≤ 10} etc.

(3) Infinite Set- A set having an infinite number of elements is said to be an infinite set. Thus,

• Set of all rational numbers between 1 and 2.
• Set of all natural numbers.
• Set of all points on a plane etc. are some examples of an infinite set.

(4) Singleton Set- A set consisting of only one element is known as a singleton set. The set { 0 } is the singleton set consisting of only one element 0 (zero). Remember that it is different from the null set { }. Examples of a few singleton sets are-

• { x : x is an even prime }.
• { 5 } etc.

(5) Equal Sets- Two sets A and B are said to be equal sets if every element of set A ∈ set B. Thus,

• A = { 1, 3, 5, 7, 9 } and B = { x : x is an odd integer less than 10 } are equal sets and we write A = B.
• Similarly, A = { x : x is the least whole number } and B = { 0 } are equal sets.

(6) Equivalent Sets- Two sets A and B are said to be equivalent if the number of elements in set A is equal to the number of elements in set B.

The number of elements of a finite set A is known as its order or cardinal number and is denoted by n (A). For two sets A and B to be equivalent n (A) = n (B).

The two sets, A = { 2, 3, 5 } and B = { 1, 2, 4 } are equivalent and we write A ↔︎ B.

(7) Subsets-

(8) Universal Set- The superset of each of the given sets is the universal set. Thus, a set that contains all the sets in a given context (i.e., a set of which all the given sets are subsets) is called the Universal set. Universal sets are usually denoted by “ξ“.

(9) Power Set- The family of all the subsets of a given set A is known as the power set of A and is denoted by P (A). If there be n elements in set A, then the number of elements in P (A) is 2ⁿ.