In this chapter we will learn about the definition of equal and equivalent sets with properties and solved examples.
Equal sets
Two sets A & B are equal when both the sets have same elements.
This means that every element of set A is a member of B and every element of set B is member of A.
Note:
The arrangement of elements in the set is not important.
We have to look if all the elements in both set are similar or not.
Representing equal sets
If two sets A & B are equal then the relationship can be expressed as;
A = B
The expression signifies that both the elements have same elements.
Examples of Equal set
(i) A = { 2, 5, 7, 9 }
B = { 7, 2, 5, 9 }
Explanation
Both the sets A & B are equal since all elements of set A are also part of set B and vice- versa.
Note that the elements in the given set can be listed in any order.
(ii) A = { x : x is letter in word ” LOYAL ” }
B = { x : x is letter in word ” ALLOY ” }
Explanation
Representing both the sets in Roster form.
A = { x : x is letter in word ” LOYAL ” }
Roster form of set A is;
A = { L, O, Y, A }
Here we have removed the repeated elements.
Roster form of set B is;
B = { A, L, O, Y }
Both the sets are equal since they contain same elements.
Hence, A = B
(iii) P = {1, 2, 3}
Q = {1, 2, 2, 3}
Check if the above two sets are equal.
Explanation
The two given sets are;
P = {1, 2, 3}
Q = {1, 2, 2, 3}
Removing the repeated elements from set Q.
Q = {1, 2, 3}
Both the sets P & Q have the same elements, hence they are equal.
P = Q
(iv) A = { x : x is a letter of word “WOLF” }
B = { x : x is a letter of word “FOLLOW” }
Solution
Writing the above sets in Roster form.
A = { W, O, L, F }
B = { F, O, L, W }
Observe that all elements of set A are also part of set B and vice-versa.
Hence, A = B.
Equivalent Set
Two sets are equivalent when their order number is same.
Hence, the equivalent sets have same number of unique elements.
Representing Equivalent set
If two sets A & B are equivalent then the relationship can be expressed as;
n (A) = n (B)
Where n(A) & n(B) tells the cardinal number of sets.
Examples of Equivalent set
(i) Given below are two sets A & B. Check if the sets are equivalent or not.
A = { 1, 2, 3 }
B = { a, b, c }
Solution
Let’s find the cardinal number of both the sets.
A = {1, 2, 3}
There are three unique elements in set A.
So the cardinal number of set A is 3.
n(A) = 3
B = { a, b, c }
There are three unique elements in set B.
So the cardinal number of set B is 3.
n(B) = 3
You can see that n(A) = n(B) = 3.
Hence both the sets A & B are equivalent set.
(ii) Given below are four sets A, B, C & D. Pair the equivalent sets.
A = { 5, 7, 10, 13 }
B = { l, m, r }
C = { x, y, z, b, z, b, x }
Solution
Let’s find the cardinal number of each of the sets.
A = { 5, 7, 10, 13 }
There are 4 unique elements in the set.
n (A) = 4
B = { l, m, r }
There are 3 unique element in set B.
n (B) = 3
C = { x, y, z, b, z, b, x }
Removing the repeating elements.
C = {x, y, z, b}
There are three unique elements in set C.
n (C) = 4
In the above three sets, set A & C have the same order.
n(A) = n(C) = 4
Hence both the sets are equivalent sets.