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102 LearnersLast updated on October 16, 2025
Equal Sets
Set theory states that two sets are equal if all of their elements are identical, regardless of order.
What are Equal Sets?
Equal sets are made up of the same elements. This means that each element in one set needs to be present in the other. For example, Sets A = {2, 4, 6, 8, 10} and B = {2, 4, 6, 8, 10} are two examples. They are equal sets since they have the same elements, despite having different orders.
Difference Between Equal and Equivalent Sets
|
Equal Sets |
Equivalent Sets |
|
In two or more sets, all the elements are equal. |
Two or more sets are equivalent if they have the same number of elements. |
| The cardinality of equal sets is the same. |
The cardinality of equivalent sets is the same. |
|
Their element in both sets is identical |
Only the number of elements is the same |
|
‘=’ is the symbol for equal sets. |
~ or ≡ is the symbol used to represent equivalent sets. |
|
All equal sets are equivalent sets. |
Equivalent sets may or may not be equal. |
How to represent an equal set in a Venn diagram?
Equal sets are represented by fully overlapping circles in a Venn diagram, since they have identical components. For example, A = B if set A = {11, 22, 33} = B = {11, 22, 33}. As a result, the two circles that stand for A and B will exactly overlap, demonstrating their content equality.
What are the Properties of Equal Sets?
Equal sets are simple to recognize and comprehend due to a few essential properties:
- The equality of the two sets is unaffected by the elements’ order.
- Equal sets have the same number of elements, or the same cardinality.
- All elements in equal sets must be equal.
Real Life Applications of Equal Sets
Let us see how equal sets help in real life.
Verification of student records
Schools compare groups of students who paid fees and those who were enrolled in a course; if the two sets are equal, it means everyone who enrolled has paid, and no unpaid students are available.
Management of inventories
To ensure accurate stock tracking, warehouses use equal sets to confirm that scanned items match those listed in the system.
Check for database duplication.
Comparing two sets of user entries helps in data cleaning by determining whether there are duplicate entries or whether the information in both sets is identical.
Monitoring attendance and submissions
All present students turned in their work if the sets of students who attended class and those who turned in homework are equal, according to the teachers.
Verification of voters
Election officials make sure to check whether no extra or missing votes were recorded by comparing the number of registered voters with the number of votes cast, where equal sets help to verify the voters.
Common Mistakes and How to Avoid Them in Equal Sets
Here are some common mistakes students make regarding equal sets with solutions.
Mistake 1
Confusing equal sets with equivalent sets
Always remember, the elements of equal sets must be identical. Only the number of elements in the equivalent set is the same. For example, {4, 5, 6} = {6, 5, 4} but {4, 5, 6} ≠ { 7, 8, 9} (they are equivalent, not equal).
Mistake 2
Thinking order matters in sets
Always know that sets are not arranged in any particular order. {6, 7, 8} = {8, 6, 7}. Always look for presence instead of position.
Mistake 3
Counting duplicate elements in a set
Remember that in sets, duplicates don’t matter. Hence, {5, 6, 6, 7} = {5, 6, 7}. The equality of sets depends on unique elements. When checking for equality, don’t count elements that are repeated.
Mistake 4
Assuming the empty set equals a set with zero
Always know that there are no elements in the empty set ∅. Zero is the only element in {0}. So, ∅ ≠ {0}. They are not the same.
Mistake 5
Comparing different data types as equal
Always verify that the element types differ. {1} ≠ {‘1’} since one is a string and the other is a number.
Solved Examples on Equal Sets
Problem 1
Do the sets A = {8, 9, 3, 4} and B = {4, 3, 8, 9} have equal values?
Yes, they have equal values. A = B.
Explanation
The order of elements is irrelevant in set theory. But, 4 elements 8, 9, 3, and 4 are in set A. These four elements are also present in set B, though the order is different. Both sets are said to be equal since they don’t contain any extra or missing elements.
Problem 2
Are A = {orange, grapes, mango} and B = {grapes, mango, orange, orange} equal?
Yes, A = B
Explanation
Sets automatically ignore repeated elements, even though set B seems to have an extra “orange”. Since every element in set theory is distinct, multiple instances of the same element are only counted once. The same unique components, grapes, mango, and orange, are present in both sets. The sets are equal as a result.
Problem 3
Are A = {0} and B = ∅ equal?
No, they are not equal, A ≠ B
Explanation
The number 0 is the only element in set A. This indicates that its cardinality, or total number of elements, is 1. The empty set, B (∅), doesn’t have any elements. A set differs even if 0 is present. These sets are therefore not equal.
Problem 4
Do the sets A = {x, y, z} and B = {v, x, y, z} have equal values?
No, they are not equal A ≠ B
Explanation
The sets are equal because they have the same elements. The three components of set A are x, y, and z. The same three components are present in set B, along with the additional element “v”. This extra component gives set B content that set A does not. Therefore, they are not an equal set.
Problem 5
Are A = {‘2’, 4} and B = {2, 4} equal?
No, they are not equal A ≠ B
Explanation
Here, the elements ‘2’ and 2 are different; one is a string and the other is an integer. Even though the number 4 appears in both sets. ‘2’ is a text string and 2 is an integer number. Data types are important in set theory; ‘2’ and 2 are not the same element. The sets are not equal because one element is different.
FAQs on Equal Sets
1.What is meant by an equal set?
2.What does the equal sets symbol mean?
3.How to represent an equality between two sets?
4.What are the equivalence sets?
5.What prerequisites must equal sets meet?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
