Equal Set

A set is a collection of related objects enclosed in brackets. Learn the distinctions and similarities between an equal set and an equivalent set, as well as how to define cardinality and the concept of equivalent sets.

Before we can define an equivalent set, we must first understand what a set is. A set is a group of items that are usually related to one another. All Sets are always enclosed within brackets: { }. A set of numbers, words, or even pictures can be created. Here are some set examples:

April, May, June, and July are the four months of the year.

{1, 2, 3, 4, 5, 6}

When a set goes on indefinitely, the last element is followed by three dots known as an ellipsis, indicating that the numbers continue. The following is an example: 1, 2, 3, 4, 5, 6…

Equal set

The Equal sets are created when two sets of elements have the exact identical elements. It makes no difference how the items in a set are organised. It’s only important that each set have the same components. Equal sets can be seen in the following examples below:

• {1,2,3,4}  and {4,3,2,1} 
• {April, May, June, July } and { July, June, May, April}

A set with an equal number of elements is called an equivalent set. The sets don’t have to be identical in every way; they just have to have the same number of elements. Consider the following examples which are given below:  

Example 1:

• Set  A : {M,N,O,P}
• Set  B: {January , February, March, April, May}

Despite the fact that Sets A and B have completely different elements (Set A contains letters and Set B has months of the year), they both have five. Set A has five letters, while Set B has five months. As a result, they’re equivalent sets! 

 Example 2:

• Set M: { Lion, Tiger, Cat, Dog}
• Set  N: {Oranges, Apples, Mangoes, Grapes}

Sets M and N both contain word items from completely different categories (Set M includes articles of Animal, while Set N includes fruit types), yet they both have the same number of elements, four. As a result, they’re equivalent sets!

Equal Set’s Symbol

The sign “=“, which stands for equality, is used to represent equal sets.

The sign “” is for unequal sets which means “not equal to.”

In a similar vein to the second example,

N = M Set M is equal to Set N, for example.

Set M, on the other hand, is not equal to Set C.

Properties of Equal Sets 

Equal sets are those in which all elements of two or more sets are equal and the number of elements is likewise equal. The symbol that represents equal sets is ‘=,’ i.e., A = B is written if sets A and B are equal. We know that in sets, the order of the elements doesn’t matter. So, if A = a, b, c, d and B = b, a, d, c, A and B are equal sets since their elements are the same, and the order of the elements has no bearing on the equality of the sets.  

Equivalent Sets 

To be equal, the set must have the same attributes. This means that elements in both sets should be connected one-to-one. When both sets are exhausted, the term “one to one correspondence” refers to the fact that for every element in set A, there exists an equal number of elements in set B.

When the cardinality of two sets A and B is the same, n (A) = n (B), they are said to be equal (B).In general, two sets can be said to be equivalent if the number of elements in each sets is the same. And they don’t have to have the same elements or be a subset of one another. 

Equal and Equivalent Sets 

 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 consequence, two sets of equal cardinality can be examined or compared.

Although analogous sets are always equal, they are not necessarily equal.

Conclusion

 A set, as previously established, is a well-defined collection of things. We already know that all null sets are comparable. If A and B, are two sets that are equal, then A is equivalent to B. It implies that two equal sets are always similar, but the reverse is not necessarily true. There aren’t any infinite sets that can be compared to each other. The sets of all real numbers and the set of integers, for example, are not equal.