Equal Sets: Definition, Cardinality, and Venn Diagram

Equal Set is the relation between two sets that tells us about the equality of two sets i.e., all the elements of both sets are the same and both sets have the same number of elements as well. As we know, a set is a well-defined collection of objects where no two objects can be the same, and sets can be empty, singleton, finite, or infinite based on the number of its elements.

Other than that, there can be sets based on the relationships between two sets such as subsets, equivalent sets, equal sets, or it can set of subsets for any set, i.e., power sets, etc. This article explores one such relationship of sets known as Equal Set, including definition, examples, properties as well as Venn diagram.

What are Equal Sets?

Equal sets are those sets whose cardinality is the same and whose all elements are equal. In other words, two sets are regarded as equal sets when they have all the same elements and also the same number of elements.

Equal Sets Definition

If all elements of two or more sets are equal and the number of elements is also equal, then the sets are said to be equal sets.

Example: P = {a, b, c, d} and Q = {a, b, c, d} are equal sets since they both have the same elements and also the same number of elements.

Equal Set Symbol

Equal Set is represented by “=” sign between any two sets, where the equality holds. For example, {2, 3, 5} and {3, 2, 5} are equal sets and we can represent them using “=” symbol as follows:

{2, 3, 5} = {3, 2, 5}

Unlike this, unequal sets are represented by “≠” which means the equality between sets doesn’t hold. For example,

{2, 3, 5} ≠ {1, 2, 3}

Example of Equal Sets

Let P be the set of all integers greater than 0 and Q be the set of all natural numbers.

Then, P = {1, 2, 3, 4,…..} and Q = {1, 2, 3, 4,….}.

As we can see, all elements of P are the same as all the elements of Q, P and Q are equal sets.

Some other examples include:

• A = { 1, 2, 3, 4, 5} and B = {2, 3, 1, 5, 4}.
• Set of alphabets in words “listen” and “silent”.
• Set of fractions {1/2, 2/4, 3/6} and {6/12, 4/8, 2/4}.

Equal and Unequal Sets

The key differences between both equal and unequal sets are as follows:

Aspect	Equal Sets	Unequal Sets
Definition	Two sets have the same elements.	Two sets have different elements.
Notation	A = B	A ≠ B
Cardinality	Equal	May or Maynot be Equal
Examples	{1, 2, 3} = {3, 2, 1}	{1, 2, 3} ≠ {4, 5, 6}
Subsets	Every subset of A is also a subset of B, and vice versa.	Subsets may differ.
Intersection	A ∩ B = A (or B)	A ∩ B has common elements of both A and B.
Union	A ∪ B = A (or B)	A ∪ B combines elements of both A and B.
Complement	Complement of A is same as complement of B.	Complements of unequal sets differs.

Equal and Equivalent Sets

The key differences between equivalent and equal sets are given in the following table:

Equal Sets	Equivalent Sets
Two or more sets are equal when all their elements are equal.	Two or more elements are equivalent when they have the same number of elements.
Equal sets are denoted by the symbol ‘=’.	Equivalent sets are denoted by the symbol ‘~’ or ‘≡’.
Equal sets is a broader term and encompasses equivalent sets, i.e., all equal sets are also equivalent sets.	Two or more equivalent sets may or may not be equal.
All elements of equal sets need to be the same.	The elements of two equivalent sets need not be the same.

Note: Equal Sets are always Equivalent Sets but vice versa is not true.

Venn Diagram of Equal Sets

The following Venn diagram shows set A = {2, 3, 5} = set B.

Read more about: Venn Diagram

Properties of Equal Sets

There are various properties of equal sets, some of which are listed as follows:

• The intersection of two equal sets is equal to both sets, i.e., if A = B then, A ∩ B = A = B.
• Two equal sets are always subsets of each other, i.e., if A ⊂ B and B ⊂ A, then A = B.
• For two sets to be equal, the order of their elements does not matter, i.e., {9, 10, 11} = {11, 10, 9}.
• The cardinality of equal sets and their power set are the same.
• Equal sets always have the same number of elements.
• The elements of two equal sets are equal.

Practise Problems of Equal Sets

P1. Determine if A = {a, b, c} and {b, c, a} are equal set or not.

P2. Check if set A = {2, 4, 6, 8} and set B = (x: x is positive even integer less then 10)

P3. Determine if the sets P = (x: x is roots of equation, x² + 5x + 6 = 0) and Q = {2, 3} are equal set or not.

Solved Example of Equal Sets

Problem 1. Are the sets P = {r: r is prime such that 40 < r < 50} and Q = {42, 44, 45, 46, 48} equal?

Solution:

Set P = {r: r is prime such that 40 < r < 50} and set Q = {42, 44, 45, 46, 48, 49}.

Thus, P = set of prime numbers between 40 and 50.

⇒ P = {41, 43, 47} ≠ {42, 44, 45, 46, 48, 49} = Q

Thus, sets P and Q are unequal.

Problem 2. Identify the equal sets from the following:

P = {p ∈ R: p²– 2p + 1 = 0}
Q = (1, 2, 3}
R = {p ∈ R : p³ – 6p² + 11p – 6 = 0}.

Solution:

Two sets are regarded as equal sets when they have all the same elements and also the same number of elements.

Let’s list out the elements of sets P and R before comparing them with set Q.

P ={p ∈ R: p² – 2p + 1 = 0}

⇒ p² – 2p + 1 = 0

⇒ (p – 1)² = 0

∴ p = 1.

⇒ P = {1}

Set Q can also be written as {1, 2, 3} since we do not repeat elements in a set.

Similarly, upon solving p³ – 6p² + 11p – 6 = 0, set R = {1, 2, 3}.

Thus, sets Q and R are equal.

Problem 3. Determine the groups of equivalent and equal sets from the following: A = {0, $}, B = {10, 21, 39, 94}, C = {44, 89, 128}, D = {39, 10, 21, 94}, E = {1, 0}, F = {89, 44, 128}, G = {15, 5, @, 11}, H = {a, c}.

Solution:

Equivalent Sets:

Having 2 elements each: A, E and H

Having 3 elements each: C and F

Having 4 elements each: B, D and G

Equal Sets:

B and D = {10, 21, 39, 94}

C and F = {44, 89, 128}

Problem 4. Determine whether the sets of alphabets in words TITLE and LITTLE are equal.

Solution:

Let A be the set of alphabets in the word TITLE.

A = {L, I, T, E}

Let B be the set of alphabets in the word LITTLE.

B = {L, I, T, E}

Thus, A and B are equal sets.

FAQs on Equal Sets

What is the Meaning of Equal Sets?

Two sets are regarded as equal sets when they have all the same elements and also the same number of elements.

Which symbol is used to represent Equal Sets?

Equal sets are represented with the notation ‘=’ between them as follows:

P = Q

How do you Prove Sets are Equal?

To proof any two given sets are equal sets we need to asses two things in given sets.

• Both sets needs to have same number of elements i.e., both sets have same cardinality.
• All the elements of each set are same as the other.

If any two sets hold the following two conditions, those are equal sets.

When are Two Sets Equal?

When both sets have equal number of elements and all the elements of one sets are same as other set, then both the given sets are equal.

What are Equal and Unequal Sets?

Equal sets are those which has equal number of same elements in both the sets, on the other hand if set either have unequal number of elements, different elements or both, then sets are said to be unequal.

What is the difference between Equal and Equivalent Sets?

Equal Sets are those sets whose all elements are equal and are represented with a “=” sign between two sets.

However, equivalent sets are those which have the same cardinality i.e., an equal number of elements and ‘~’ or ‘≡’ is used between two sets to represent this equivalent relation between them.

What are some Properties of Equal Sets?

Some porperties of Equal Sets are:

• If A = B, then A ⊂ B and B ⊂ A.
• If A = B then, A ∩ B = A = B.

“All Equivalent Sets are Equal Sets” is True or False.

No, all equivalent sets are not equal as equivalent sets can have different element in each other but equal sets can’t have that.

What is an Example of Equal and Unequal sets?

For sets A = {a, e, i, o, u}, B = {a, i, o, u, e} and C = {a, 1, e, 5, i}

A = B but A ≠ C and B ≠ C as well.

Thus, A and B are examples of Equal Sets, and A & C or B & C are examples of Unequal Sets.