Set elements || Understand the elements of set with examples

In this post we will learn about elements or objects of sets with solved examples.

What are set elements?

The objects or members present inside the set is called set elements.

Set elements are present inside the curly bracket that represent any particular set.

Examples of set elements

(i) E = { 2, 4, 6, 8, 10 }

Here;
E is the set name.
2, 4, 6, 8 and 10 are the set elements.

(ii) Y = { “Red”, “Orange”, “Yellow”, “Green”}

Here,
Y is the set name.
Red, Orange, Yellow and Green are the set elements.

(iii) A = { -3, 0, 9, 15}

Here A is the set name.
-3, 0, 9 and 5 are the set elements.

Representing set elements

We use the symbol epsilon ” \mathtt{\ \epsilon \ } ” to show that the given element is the part of the set.

For Example, consider the below set.

X = { 9, 19, 25, 30 }

Here;
X is the name of the set.
9, 19, 25 and 30 are the elements of set.

We can say that;
9 \mathtt{\ \epsilon \ } X, It means element 9 belongs to X.

Similarly we can write this for all the elements of X.
19 \mathtt{\ \epsilon \ } X ;
25 \mathtt{\ \epsilon \ } X ;
30 \mathtt{\ \epsilon \ } X ;

Remember the significance of above symbol as it is directly asked in the examinations.

What does symbol \mathtt{\notin } mean in set theory?

The symbol \mathtt{\notin } represents that the element doesn’t belongs to a particular set.

For Example, consider the below set.
X = { 5, 19, 25, 54}

We know that number 60 doesn’t belongs to the set X.
This statement can be expressed as: 60 \mathtt{\notin } X.

What is size of a set?

The number of elements present in a given set is known as size of a set.

For Example, consider the below set;
A = { 19, 23, 51, 64, 71 }

Note that there are 5 elements in the set; 19, 23, 51, 64 and 71.
Hence, the size of the set is 5.

Similarly, consider the below set Y’
Y = { 2, 4, 6, 8, 10, 12 . . . . . 2n}

Here Y is the never ending set of even numbers.
The number of elements in set Y is infinite.
Hence, the size of set is also infinite.

Solved examples of elements of Set

(01) Given below is the set A.
A = { 99, -33, 21, 15, -73, 21 }
List down all the elements/members of the set.

Solution
The elements are set are: 99, -33, 21, 15, -73 and 21.

(02) List down all the elements of the below set X
X = { A, P, L, E }

Solution
The elements of the set are: A, P, L, E

(03) Given below are statements regarding set A. Check is the statement is True or False.

A = { -2, 6, -5, 10, -15, 20 }

(i) 6 \mathtt{\ \epsilon \ } A
(ii) 5 \mathtt{\ \epsilon \ } A
(iii) 20 \mathtt{\ \epsilon \ } A
(iv) -15 \mathtt{\ \epsilon \ } A
(v) 7 \mathtt{\ \epsilon \ } A

Solution

(i) 6 \mathtt{\ \epsilon \ } A
The statement is correct as 6 is an element of set A.

(ii) 5 \mathtt{\ \epsilon \ } A
The statement is incorrect as 5 is not an element of A.

(iii) 20 \mathtt{\ \epsilon \ } A
It’s a correct statement as element 20 belongs to set A.

(iv) -15 \mathtt{\ \epsilon \ } A
It’s a correct statement.

(v) 7 \mathtt{\ \epsilon \ } A
This statement is False.

(04) Given below is the set Y.
Y = { \mathtt{\frac{1}{2} ,\ \frac{1}{6} ,\ \frac{1}{18} ,\ \frac{1}{36}} }

Check if the below statements are correct or not.

(i) \mathtt{\frac{1}{2} \ \notin \ Y}
(ii) \mathtt{\frac{1}{3} \ \epsilon \ Y}
(iii) \mathtt{\frac{1}{18} \ \epsilon \ Y}
(iv) \mathtt{\frac{1}{7} \ \notin \ Y}
(v) \mathtt{\frac{1}{36} \ \epsilon \ Y}

Solution

(i) \mathtt{\frac{1}{2} \ \notin \ Y}

This is incorrect statement.
1/2 is an elements belongs to set A.
So the correct expression is; \mathtt{\frac{1}{2} \ \epsilon \ Y}

(ii) \mathtt{\frac{1}{3} \ \epsilon \ Y}
This is an incorrect statement.
1/3 is not the member of set Y.

(iii) \mathtt{\frac{1}{18} \ \epsilon \ Y} .
This is a correct expression.
Element 1/3 belongs to set Y.

(iv) \mathtt{\frac{1}{7} \ \notin \ Y}
This is a correct expression.
1/7 is not part of set Y.

(v) \mathtt{\frac{1}{36} \ \epsilon \ Y}
This is correct statement.
Element 1/36 belongs to set Y.

(05) Find the size of the below sets.

(i) A = { 2, 9, -13, 7, 13}
(ii) B = { 23, 19, 7 }
(iii) X = { A, M, U, V }
(iv) Y = { 0 }
(v) Z = { }

Solution
(i) A = { 2, 9, -13, 7, 13}
There are 5 elements in the given set, namely 2, 9, -13, 7, 13.
Hence, the size of set is 5.

(ii) B = { 23, 19, 7 }
There are 3 elements in the set.
Hence, the size of set is 3.

(iii) X = { A, M, U, V }
There are 4 elements in set, namely A, M, U , V.
Hence, the size of set is 4.

(iv) Y = { 0 }
There is only one element in set, namely 0
Hence, the size of set is 1.

(v) Z = { }
There is no element is set Z.
Hence, it is a null set.