Equal and Equivalent Sets Clear Explanation with Examples and FAQs

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A set is defined as a collection of well-defined separate objects in mathematics. The elements of a set are the various objects that make up the set. The set's elements can be written in any sequence, but they should not be repeated. 

The capital letter is frequently used to denote the set. Two sets can be equivalent, equal, or unequal to each other in basic set theory. In this article, we'll look at what equal and equivalent sets are and the differences between them, using examples. 

Basically, we will be answering your question, "how do you find equal sets?" 

Define Equal Sets  

Equal Sets are a type of set that has the same number of items in it. Only if each element of set A is also an element of set B can two sets, A and B, be equal. Also, two sets are said to be equal if they are subsets of each other. This is represented by the following: 

A = B 

A ⊂ B and B ⊂ A ⟺ A = B 

The sets are said to be unequal if the condition described above is not met. This is represented by the following: 

A and B 

Let's see if we can figure out when the two sets are equivalent. 

Define the terms "equivalent sets"  

In mathematics, equivalent sets have two definitions. 

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

Definition 2 of Equivalent Sets: Assume that two sets A and B are equivalent only if they have the same cardinality, n(A) = n(B) (B).

As a result, the sets must have the same cardinality to remain comparable. This condition requires a one-to-one relationship between the components of both sets. 

In this case, the one to one requirement means that for every element in set A, there are equal elements in set B until both sets A and B are exhausted. 

As a result, if the number of elements in both sets remains equal, the two sets are identical in general. The sets don't have to have the same elements to be a subset of one another. 

Example of an Equal Set

If P= {1, 3, 9, 5, −7} and Q = {5, −7, 3, 1, 9,}, then it can be stated that P = Q. It's worth noting that no matter how many times an element appears in a set, it is only counted once. It's also worth noting that the order of the components in a given set doesn't matter. 

As a result, equal sets can be described in terms of cardinal number:  

If C = D, then n(C) = n(D) and for any x ∈ C, x ∈ D too. 

Example of an Equivalent Set 

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

C is identical to D if C = {x: x is a positive integer} and D = {d: d is a natural number}. 

Points to remember: 

All the null sets are equivalent to each other.

A is equivalent to B if A and B are two sets where A = B. This means that two equal sets are always comparable, although the opposite isn't always the case. 

Certain infinite sets aren't comparable to each other. Consider the sets of all real numbers and integers, for example. 

Conclusion

Equal sets have the same precise components, even if they are not in the same order. Equivalent sets are made up of distinct items but the same number of them. The number of elements in a set is known as its cardinality. 

As a result, two sets with the same cardinality are comparable. Although equal sets are equivalent, equivalent sets are not always equal.