Equivalent Sets

Set Theory is a branch of mathematics where we learn different types of sets and their properties. In Maths, sets are defined as a collection of well-defined objects or elements. These objects are also known as elements or members of a set. Sets are represented in two forms i.e set-builder form or roster form. A set is represented by a capital letter.  The number of elements in the finite set is called the cardinal number of a set and represented in a curly bracket {…}. Suppose, set A is a collection of all the natural numbers. It is represented as A = {1,2,3,4,5,6,7,8,…..∞}.

In set theory, two sets can either be equivalent, equal or unequal to each other. In this article, we will define equal sets, what is meant by equal and equivalent sets with examples and also the difference between them.

Equivalent Set Definition

Two sets are said to be equivalent if their cardinality number is the same. This means that there must be one to one correspondence between elements of both sets. Here, one to one correspondence means that for each element in set A, there exists an element in set B until sets get exhausted.

If two sets A and B have the same cardinality then there exists an objective function from set A to B.

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

Generally, we can say that two sets are equivalent to each other if the number of elements in both sets is equal. It is not necessary that both the set have similar elements, or they are a subset of each other.

Equal Set

Equal Set Definition - Two sets A and B are said to be equal only if each element of set A is also present in an element of the set B. In another way, we can say if two sets are the subsets of each other, they are said to be equal. It is represented by:

A = B

A ⊂ B and A ⊂ B ⇔ A = B

If it doesn’t satisfy the above condition, then the sets are said to be unequal. It is represented by:

A ≠ B

Equal Set Example

If A = {1, 3, 8, -2, −7} and B = {-2, −7, 3, 1, 8,}, then A = B. We can note that no matter how many times an element is repeated in the set, it is counted only once. Also, the order of elements doesn’t matter in a set. So, if we represent in terms of cardinal number, we can say that:

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

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In the above diagram, these are the two sets of different shapes. These are equal sets because the number of elements is the same and their elements are also the same.

Symbol of Equal Set

Equal sets are represented by a symbol of  “=” i.e. equality.

Unequal sets are represented by the symbol of “≠” i.e. not equal to.

As in the above example,

A = B i.e. Set A is equal to Set B

But, A ≠ C i.e. Set A is not equal to Set C.

Equivalent Set Example

If P = {1,−7,200, 90,55} and Q = {1,2,3,4,5}, 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.

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In the above diagram, all the four sets are equivalent sets because the number of elements is the same in all the four sets of triangle, smiley, star and heart. Here n(Triangle) = n(Smiley) = n(Stars) = n(Heart) = 6.

Equivalent Sets Symbol

Equivalent sets are represented by an equivalent sign  or “ ~” or “≡”

As in the above example, P ~ Q i.e. Set P is equivalent to Set Q. 

Following are the difference between Equal and Equivalent Sets :

Difference Between Equal and Equivalent Sets

Equal Sets and Equivalent Sets

Relation Between Equal and Equivalent Sets

Two sets are equal if they have exactly the same element because their elements and the number of elements both are the same without any order and repetition of elements.

Two sets are equivalent if their number of elements are the same. These elements can be the same or different, but the number should be the same.

Equal sets are equivalent also

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In the above diagram, we can see both the sets are equal because they’re all the elements are the same and their number of elements are also the same so these are equivalent also.

So all equal sets are equivalent also.

Equivalent sets may not be equal sets

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Here X and Y are equivalent sets as a number of elements are the same i.e. 4, but these are not equal sets because elements of both sets are not the same.

So we can say all equivalent sets are not equal sets.

Solved Problems:

Classify the Following Sets as Equivalent or Equal Sets?

(i) A = It contains a set of vowels in the English alphabet.

B = It contains a set of all letters in the word “VOWEL”

(ii) E = The set E={ x : x is a letter in the word “LIFE”}

F = {F, I, L, E}

Sol : Consider set A and B

A  =  {a, e, i o, u}

B  =  {V, O, W, E, L}

n(A)  =  5, n(B)  =  5

Since n(A)  =  n(B), they are equivalent sets.  

(ii) consider set E and F

E  =  { L, I, F, E }

F = {F, I, L, E}

In both sets, we have the same elements. Hence they are equal sets.

Conclusion

As we have discussed, set is a well-defined collection of objects . As we know all the null sets are equivalent to each other. Consider A and B are two sets such that A = B, then A is equivalent to B. It means two equal sets will always be equivalent but the converse of this statement is not always true. All infinite sets are not equivalent to each other. For example, the set of all real numbers and the set of integers are not equivalent to each other.