A Is Equivalent To B

In Mathematics, a set is defined every bit the collection of well-defined distinct objects. The unlike objects that create a set are called the elements of the set up. By and large, the elements of the sets can be written in any society but it should non exist repeated. The set is usually represented past the upper-case letter. In basic gear up theory, 2 sets can either exist equivalent, equal or unequal to each other. In this article, nosotros 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 tin be equal only if each element of set A is also the element of the set B. As well 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 status discussed above is not met, then the sets are said to exist unequal. This is represented by:

A ≠ B

Let united states now go ahead and detect when the given 2 sets are equivalent.

What are Equivalent Sets?

To exist equivalent, the sets should have the same cardinality. This ways that at that place should be one to one correspondence between elements of both the sets. Hither, 1 to one correspondence means that for each chemical element in the fix A , there exists an element in the set B till the sets become 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 ) = northward ( B ) .

In general, we tin 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 = { ane , 3 , ix , five , − 7 } and Q = { 5 , − 7 , 3 , ane , ix , }, and then P = Q . It is also noted that no matter how many times an chemical element is repeated in the fix, it is just counted in one case. Also, the social club doesn't matter for the elements in a set. And then, to rephrase in terms of cardinal number, nosotros can say that:

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

Equivalent Ready Example

If P = { 1 , − 7 , 200011000 , 55 } and Q = { one , ii , iii , 4 }, then P is equivalent to Q .

If C = { x : 10 i due south positive i north t e thousand e r } and D = { d : d i south a n a t u r a l n u m b e r }, then C i south e q u i five a l e north t t o D .

Important points:

All the null sets are equivalent to each other.
If A and B are two sets such that A = B , and so 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 existent numbers and the ready of integers.

Video Lesson on What are Sets

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