Equal And Equivalent Sets

Equal sets are sets in set theory that have the same number of elements and are all equal. It’s a set equality idea. Let us review the definition of sets before delving more into the concept of equal sets. A set is a well-defined group of objects like letters, numbers, persons, forms, and so on. In set theory, we look at several forms of sets. We’ll look at the concept of equal sets, its definition, and its features in this article. We’ll also use examples to further understand the distinction between equal and equivalent sets.

Definition of Equal Sets: When all of the items in two or more sets are the same and the number of elements is also the same, the sets are said to be equal sets. Equal sets are denoted by the symbol ‘=,’ i.e., if sets A and B are equal, A = B is written. 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} , then A and B are equal sets since their elements are the same, and the order of the components has no bearing on the equality of the sets.

Definition Of Equal Equivalent Sets: The sets must have the same cardinality to be comparable. This implies that elements from both sets should have a one-to-one connection. One to one correspondence means that for every element in set A, there is an equal number of elements in set B until the sets are exhausted.

What are Equal Sets?

Equal sets are defined as sets with the same cardinality and all equal elements. In other words, if two or more sets have the same items and the same number of elements, they are said to be equal sets. Set A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5} as an example. Sets A and B are thus considered to be equal sets because their elements are the same and their cardinality is the same. If all of the elements in two sets are not the same, they are said to be unequal sets, while sets with the same number of elements are called equivalent sets. For instance, suppose A = {1, 2, 3, 4, 5) C = {2, 4, 6, 7, 9} and D = {2, 5, 6}. Sets A and C both have the same number of elements, but not all of them are the same. As a result, A and C are the same set. Now, the cardinality of sets A and D is not the same, and the elements are not equal. As a result, sets A and D are unequal. The number of elements and similarity of elements in the two sets can be used to understand the equal and equivalent sets.

Properties of Equal Sets

We’ve now grasped the concept of equal sets. Following that, we’ll look at some of their key characteristics that aid in recognising and identifying them:

• The equality of the two sets is unaffected by the order of the elements.
• Equal sets have the same cardinality, or number of elements, as of each other.
• All elements in equal sets must be equal.
• The cardinal number of the power set of equal sets is also the same.
• The property of equal and equivalent sets is that they have the same amount of elements.
• The set notation used when two sets are subsets of each other is A⊆B and B⊆A, and the two sets are equal. A is equal to B.
• All equal sets are equivalent sets, but not the other way around.

Properties of Equal Equivalent Sets

• The equality of the two sets may or may not be affected by the order of the elements.

• If the number of elements in two or more sets is the same, then they are equal.
• The cardinality of equivalent sets is the same.
• Equivalent sets have the same cardinality.
• The symbol for similar sets is ~ or ≡.
• Equivalent sets can be equal or not.
• Equivalent sets may or may not be equivalent.

Difference Between Equal and Equivalent Sets

Equal Sets

• They are equal if all elements are equal in two or more sets.
• The cardinality of equal sets is the same.
• There are the same amount of elements in both of them.
• The sign ‘=’ is used to denote equal sets.
• Equivalent sets are all equal sets.
• The same elements should be used.

Equal Equivalent Sets

• If the number of elements in two or more sets is the same, then they are equal.
• The cardinality of equivalent sets is the same.
• Equivalent sets have the same cardinality.
• Alternatively, the symbol for similar sets is ~ or ≡.
• Equivalent sets can be equal or not.
• Equivalent sets may or may not be equivalent.

Important Properties of Equal Sets

• Equivalent sets do not have to be equal to be equivalent.
• Sets that have the same items are considered equal.
• If two sets are subsets of each other, they are equivalent.

Conclusion

When all of the items in two or more sets are the same and the number of elements is also the same, the sets are said to be equal sets. Equal sets are denoted by the symbol ‘=,’ i.e., if sets A and B are equal, A = B is written. 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} , then A and B are equal sets since their elements are the same, and the order of the components has no bearing on the equality of the sets.

The sets must have the same cardinality to be comparable. This implies that elements from both sets should have a one-to-one connection. One to one correspondence means that for every element in set A, there is an equal number of elements in set B until the sets are exhausted.