Equal and Equivalent Sets

In set theory, two or more sets can be equal, unequal, or equivalent. Understanding the differences between equal and equivalent sets is essential for solving problems in algebra, statistics, and data analysis involving set theory.

Equal Sets

Two or more sets are considered equal if they contain exactly the same elements in any order. Thus, if two sets are equal, say A and B, every element of set A is also present in set B and vice-versa.

Two equal sets, A and B, are represented by A = B

Examples

A = {c, d, x, y} and B = {d, x, y, c} are equal sets since they contain the same number of elements and the same elements arranged differently. 

A = B ⇒ {c, d, x, y} = {d, x, y, c}

Similarly, V = {2, 4, 6, 8} and W = {6, 2, 8, 4} are also equal sets.

In contrast, sets are unequal if their elements are not the same. If P = {1, 2, 4, 6} and Q = {8, 9, 10, 11, 12} are thus unequal sets,  which is represented by the symbol P ≠ Q

Properties

• All elements are equal
• They are equal regardless of the order of the elements
• They have the same cardinality
• The power set of equal sets has the same cardinality
• The set notations used when two sets are equal and are subsets of each other are: A ⊆ B and B ⊆ A ⟺ A = B

Venn Diagram

If we represent the equal sets A = {c, d, x, y} and B = {d, x, y, c} in a Venn diagram, we get

Equivalent Sets

Two or more sets are said to be equivalent if they have the same number of elements, regardless of what the elements are. Thus, two equivalent sets have the same cardinality, which means the elements of both sets correspond to each other on a one-to-one basis.

If A and C are two sets such that n(A) = n(C), they are equivalent and are represented as A ~ C or A ≡ C

Examples

A = {c, d, x, y} and B = {d, x, y, c} have the same elements (cardinality 4) written in different order. Thus, A and B are equivalent sets.

Similarly, A = {c, d, x, y} and C = {w, y, u, v}  are also equivalent sets despite them having different elements, as they have the same cardinality. 

In general, two equivalent sets need not have the same elements or be subsets of each other.

Properties

• They have the same cardinality
• The elements may or may not be arranged in a specific order
• The power set of equivalent sets has the same cardinality
• All equivalent sets are not equal sets, but the converse is true

Venn Diagram

If we represent the equivalent sets A = {c, d, x, y} and C = {w, y, u, v} in a Venn diagram, we get

Equal vs. Equivalent Sets

Factors	Equal Sets	Equivalent Sets
Definition	Two or more sets are equal if they contain exactly the same elements in any order	Two or more sets are equivalent if they have the same number of elements, regardless of the elements.
Cardinality	They have the same cardinality	They have the same cardinality
Symbol	‘=’ is used to represent equal sets	‘~’ or ‘≡’ is used to represent equivalent sets
Elements	All the elements should be the same	The elements need not be the same
Relation	All equal sets are equivalent	Equivalent sets may or may not be equal
Example	A = {1, 2, 3, 4} and B = {3, 1, 2, 4}	A = {1, 2, 3, 4} and C  = {a, b, c, d}

Solved Examples