Equal and Equivalent Sets

Even though equal and equivalent sets sound like there isn’t much difference between them. These two are similar concepts, yes, but there is a minor difference between them that sets them both apart. 

But before we divide into equal and equivalent sets, let us understand what cardinality is. Cardinality is the number of elements inside a set. Now this is important because this will help us understand the difference between equal and equivalent sets. 

Equal and equivalent sets are terms used to denote some kind of relationship between two sets. You may think of this as some sort of comparison. Like how you would compare apples to oranges but if there is no standard by which we can compare them, then it would be very difficult to establish anything. If we were to compare them by number, then we could say that there are more apples than oranges or vice versa. Or we could say that there are equal numbers of apples and oranges. 

The same way, if we were to compare two sets, we could use cardinality as a standard for comparison. 

Let us see how it is done.

Define Equal Sets

To understand Equal Set meaning, Equal Set is defined as two sets having the same elements. Two sets A and B can be equal only on the condition that each element of set A is also the element of set B. Also, if two sets happen to be the subsets of each other, then they are stated to be equal sets.

Continuing our above example, if we were to compare one basket of oranges with another basket of oranges, and if the number of oranges is equal in both the baskets, then this is said to be an example for equal sets. 

Equal Sets

An equal set can be represented by:

P = Q

P ⊂ Q and Q ⊂ P ⟺ P equals to Q

It is to be noted that if the condition discussed above is not met, then the set is stated to be unequal.

To elaborate, if the two baskets contained an unequal number of oranges or if one basket contained apples and the other contained oranges of the same number, then these cases are said to be examples for unequal sets.

Unequal sets are represented by

P ≠ Q

Define Equivalent Sets

Equivalent sets meaning in Mathematics holds two definitions.

Equivalent Sets Definition 1 - Let's say that two sets A and B have the same cardinality, then, there exists an objective function from set A to B.

Equivalent Sets Definition 2 - Let's say that two sets A and B are stated to be equivalent only if they have the same cardinality, that is, n(A) = n(B).

Thus, to remain or be equivalent, the sets should possess the same cardinality. 

In other words, if there is a basket of apples and a basket of oranges, then if they are of the same number, we can call these as an example for equivalent sets.

This condition means that there should be one to one correspondence between the elements belonging to both the sets. In this context, the one to one condition implies that for each element on the set A, there exists an element in the set B, till both the set A and set B gets exhausted.

Therefore, in general, it can be stated that the two sets remain equivalent to each other if only the number of elements in both the sets remain equal. The sets don't need to hold the same elements, or they stay to be a subset of each other.

Equal And Equivalent Sets Examples

Equal Set Example

If we consider numbers to denote the elements of two sets then we can understand equal and equivalent sets in the following manner.

Let’s understand equal sets with an example,

If M= {1, 3, 9, 5, −7} and N = {5, −7, 3, 1, 9,}, then it can be stated that M = N. It is to be noted that no matter how many times an element is repeated in a particular set, the element is counted only once. Also, it is to be pointed out that the order does not matter for the elements for a specific set. Therefore, in terms of cardinal number, equal sets can be stated that: 

If P = Q, then n(P) = n(Q) and for any x ∈ P, x ∈ Q too.

Equivalent Set Example

If S = {x: x, where x is stated to be a positive integer} and T = {d : d, where x is said to be a natural number}, then S is stated to be equivalent to T.

Thus, it can be stated that an equivalent set is simply a set with an equal number of elements. However, the sets don't need to have the same elements but must comprise the same number of elements. 

Let’s Understand Equivalent Sets With Examples

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

2. If Set G: {Sweater, Mittens, Scarf, Jacket} and Set H: {Apples, Bananas, Peaches, Grapes}, it can be noted that both Set G and Set H comprise word elements in different categories and have the same number of elements i.e. four. 

We are now clear on what equal and equivalent sets are. Now let us expand our knowledge to accommodate a few fascinating facts about the relation between equal and equivalent sets. They are mentioned as important pointers below.

Important Points to Remember on Equivalent Sets

• All the null sets are said to be equivalent to each other.

• Not all the infinite sets remain equivalent to each other. For example, the equivalent set of all the real numbers and the equivalent set of the integers. 

• If P and Q are stated to be two sets such that P is equal to Q, that is, (P = Q). This example means that two equal sets will always remain to be equivalent, but the converse of the equivalent set may or may not remain true.

• An equal set can be an equivalent set, but it is not necessary for an equivalent set to be an equal set.