Equivalent Set (original) (raw)

Last Updated : 23 Jul, 2025

An **Equivalent Set is a mathematical concept referring to two sets that have the same cardinality, which means they contain the same number of elements. The elements themselves may differ, but the sets are considered equivalent as long as their sizes are the same.

Imagine you're comparing two containers of objects, one filled with apples and another with oranges. If both containers have 1000 items, you might not care whether the items are apples or oranges; you only care that both containers hold the same quantity. This abstraction is captured by the idea of equivalent sets.

Let's consider some examples for the same:

**Example 1:

Set A = {1, 2, 3}
Set B = {a, b, c}

Since both A and B have 3 elements, they are equivalent.

**Example 2:

Set C = {x, y}
Set D = {apple, banana}

Since both C and D have 2 elements, they are equivalent.

Properties of Equivalent Sets

Important properties of equivalent sets are:

**Cardinality: If two sets have the same number of elements, they are equivalent.
**No Need for Identical Elements: The specific elements of the sets do not matter, only the count.
**One-to-One Correspondence: If you can establish a one-to-one correspondence (bijective mapping) between the elements of two sets, the sets are equivalent.

Difference between Equal and Equivalent Set

Key differences between equal and equivalent sets are:

**Aspect	**Equal Set	**Equivalent Set
**Definition	Two sets are equal if they have the exact same elements.	Two sets are equivalent if they have the same number of elements.
**Element Comparison	Elements in both sets must be identical, regardless of order.	Elements are not compared, only the count matters.
**Symbol	Denoted as A = B.	Denoted as A ∼ B.
**Example of Sets	A = {1, 2, 3}, _B = {1, 2, 3} → A = B	A = {1, 2, 3}, B = {a, b, c} → A∼B
**Focus	Focuses on the elements themselves.	Focuses on the cardinality (number of elements).
**Relationship	All equal sets are equivalent sets.	Not all equivalent sets are equal sets.

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