What is a Set?
Set definition: Each set is defined as a group of various items sharing some common trait among themselves. Set members or elements refer to the items that comprise a set. Examples of sets include a collection of cards, a group of automobiles, a group of birds, a group of days of the week, etc. Also, there exist some universally accepted sets like the set of natural numbers, the set of rational numbers, the set of real numbers, the set of integers, the set of whole numbers, the set of irrational numbers, etc.
What is the Cardinality of Sets?
The cardinality of a set is the measure of the size of the set, indicating the number of items contained in that set. It can have both finite and infinite values. The cardinality of the set A = (1, 2, 3, 4, 5, 6) is equal to 6 as set A encompasses six items. Often the cardinality of any set is referred to by the modulus sign appearing on both sides of the set name, for example, |A|.
What are the Methods of Representation of Sets?
Irrespective of the way of representation, every set is named with a capital or an uppercase English alphabet. The set names can be represented, for instance, with the letters A, B, C, D, etc. Small hand or lowercase English alphabets or numeric and other symbols represent the constituent members or the elements of sets. For instance, A = {11, 12, 13, 14, 15}, B={x: x a,c,r,t,f,h,j}
Set representations can be of the following three forms.
Set Builder Form for Representation of Sets
A definite way of describing a set making use of a single element and the statements that illustrate the characteristics of its components, separated by the symbol “:”.
To indicate whether an element is present in the given set or not, we use the Greek letter aphsilon "" . It indicates the phrase “belongs to”.
Again the symbol “∉” is interpreted as "not belonging to."
For example $A=\{x:x\text{ }is\text{ }an\text{ }even\text{ }number,\text{ }x\in N\text{ }and\text{ }x\le 25\}$
Therefore, the set A consists of the elements given by A= {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}.
Roster Form for Representation of Sets
Roster form representation refers to the method of expressing a set where the elements of the sets are discretely shown and enclosed within curly braces. The tabular form is another name for this type of set representation.
For example, N= {1,2,3,4,5,6,7,8,9,10,........}
Statement Form for Representation of Sets
Statement form representation refers to the method of denoting a set where the elements of the sets are never discretely shown, but onlt the statement is enclosed within curly braces.
For example, N= {the natural numbers}
What are the Different Types of Sets?
Some of the noteworthy various type so f sets are as follows:
Null set
This defined set contains no elements in it, and hence, is also called an empty set.
Universal Set
A set which contains all the sets defined in a mathematical statemement or problem.
Finite Set
The set whose cardinality is finite.
Infinite Set
The set whose cardinality is infinite.
Superset
By comparing any two sets A and B, if we find every element of the set B is present in set A but every element of set A is not present in the set B, we say that B is a superset of A.
Subset
By comparing any two sets A and B, if we find every element of the set B is present in the set A, we say that B is a subset of A.
Equivalent Set
By comparing any two sets A and B, if we find every element of the set B is present in the set A and vice versa, we say that B is an equivalent set of A and vice versa.
Common Sets of Numbers
The following notations can be used to represent the number sets:
N = Set of Natural numbers
W = Set of Whole numbers
R = Set of real numbers
Q = Set of Rational numbers
Z = Set of Integers
T = Set of Irrational Numbers
What is the Union of Sets?
The union of any two sets namely set P and set Q indicates the elements that are present in either of the sets P and Q. It is indicated symbolically by the statement $P\cup B$.
What is the Intersection of Sets?
The intersection of any two sets, namely set P and set Q indicates the elements that are present in both the sets P and Q. It is indicated symbolically by the statement $P\cap B$
What is a Venn Diagram?
Visual diagrams may be used to depict sets as well. A Venn diagram is the visual representation of a set. In a Venn diagram, the sets are represented graphically or visually as circles, and the intersection of the circles reveals each set's connection. The rectangular region around the sets or circles is referred to as the universal set, while the area inside the circles, or the sets, is referred to as the elements.
Venn diagram representing two sets
There are two sets A and B that overlap one another, as seen in the image above. The items shared by the two sets supplied are shown in this intersection. The universal set, denoted by the letter U, which symbolises all the items that may or may not be present in the sets, is the rectangular area around the sets.
Solved Examples
1. Write the following set statement in both set-builder form and roster form. Set of all natural numbers that lie between 1 and 40 that are prime.
Ans. the above set statement can be represented in following ways:
Set-builder form: A=x| x is prime, 1x40 and xN
Roster form: A=2,3,5,7,11,13,17,19,23,29,31,37
2. Draw the venn diagram for the following:
In a college, 200 students are selected. 120 like tea, 140 like coffee and 50 like both tea and coffee.
Ans. Let T = set of students who like tea
C= set of students who like coffee
n(T) = Cardinality of set T
= 120
n(C) = Cardinality of set C
= 140
n(T\[\cap\]C) = cardinality of students who like both tea and coffee
= 50
Students who like both tea and coffee
3. Write the set builder Form for A={2,3,4,5,6,7,8,9}
Ans. A={x| x is a natural number and 2x9}
4. Write the following set statement in set-builder form. Set of all whole numbers less than 20 that are composite.
Ans. The above set statement can be represented in following ways:
Set-builder form: $A=\{x:x\text{ }is\text{ }composite,\text{ }0\le x\le 20\text{ }and\text{ }x\in W\}$
5. Write the following set statement in roster form. Set of all whole numbers less than 20 that are composite.
Ans. The above-set statement can be represented in the following ways:
Roster form: $A=A=4, 6,8,9,10,12,14,15,16,18,20$
Summary
The sets represent the statements that can define the relationship among the elements given in a group.
The sets can be displayed in three ways: set-builder, statement form and roster form.
The cardinality of sets indicates the number of elements in a set, which can be infinite or finite.
The visual representation of sets makes use of Venn diagrams to show the correct relationship among the sets.
Practice Problems
Q1. Write the following set statement in both set-builder form and roster form. Set of all even prime numbers.
Q2. Write the following set statement in both set-builder form and roster form. Set of all odd prime numbers less than 100.
Q3. Write the following set statement in both set-builder form and roster form. Set of all even composite numbers between 200 and 300.
List of Related Articles
FAQs on Sets and Their Representation
1. What is the formula for the number of elements obtained from the union of two sets, X and Y?
The required formula for the number of elements obtained from the union of two sets X and Y is $n\left( X\cup Y \right)=n\left( X \right)+n\left( Y \right)-n\left( X\cap Y \right)$.
2. What is the formula for the number of elements obtained from the intersection of two non-overlapping sets, X and Y?
The required formula for the number of elements obtained from the intersection of two non-overlapping sets X and Y is an empty set as shown by $n\left( X\cap Y \right)=\phi $.
3. What is the formula for the number of elements obtained from the union of two non-overlapping sets, X and Y?
The required formula for the number of elements obtained from the union of two non-overlapping sets X and Y is $n\left( X\cup Y \right)=n\left( X \right)+n\left( Y \right)$
