Equal and Equivalent Sets

In Mathematics, a set is defined as the collection of well-defined distinct objects. The different objects that create a set are called the elements of the set. Generally, the elements of the sets can be written in any order but it should not be repeated. The set is usually represented by the capital letter. In basic set theory, two sets can either be equivalent, equal or unequal to each other. In this article, we are going to discuss what is meant by equal and equivalent set with examples and also the difference between them.

What are Equal Sets?

Two sets A and B can be equal only if each element of set A is also the element of the set B. Also if two sets are the subsets of each other, they are said to be equal. This is represented by:

A = B

A ⊂ B and B ⊂ A ⟺ A = B

If the condition discussed above is not met, then the sets are said to be unequal. This is represented by:

A ≠ B

Let us now go ahead and find when the given two sets are equivalent.

What are Equivalent Sets?

To be equivalent, the sets should have the same cardinality. This means that there should be one to one correspondence between elements of both the sets. Here, one to one correspondence means that for each element in the set A, there exists an element in the set B till the sets get exhausted.

Definition 1: If two sets A and B have the same cardinality if there exists an objective function from set A to B.

Definition 2: Two sets A and B are said to be equivalent if they have the same cardinality i.e. n(A) = n(B).

In general, we can say, two sets are equivalent to each other if the number of elements in both the sets is equal. And it is not necessary that they have same elements, or they are a subset of each other.

Equal And Equivalent Sets Examples

Equal Set Example

If P = {1, 3, 9, 5, −7} and Q = {5, −7, 3, 1, 9,}, then P = Q. It is also noted that no matter how many times an element is repeated in the set, it is only counted once. Also, the order doesn’t matter for the elements in a set. So, to rephrase in terms of cardinal number, we can say that:

If A = B, then n(A) = n(B) and for any x ∈ A, x ∈ B too.

Equivalent Set Example

If P = {1,−7,200011000,55} and Q = {1,2,3,4}, then P is equivalent to Q.

If C = {x : x is positive integer} and D = {d : d is a natural number}, then C is equivalent to D.

Important points:

• All the null sets are equivalent to each other.
• If A and B are two sets such that A = B, then A is equivalent to B. This means that two equal sets will always be equivalent but the converse of the same may or may not be true.
• Not all infinite sets are equivalent to each other. For e.g. the set of all real numbers and the set of integers.

Video Lesson on What are Sets

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