## What is Equal and Equivalent Sets: Introduction

## FAQs on Difference Between Equal and Equivalent Sets

1. How do we determine if two sets are equal?

To determine if two sets are equal, we compare their elements and ensure that every element in one set is also present in the other set, and vice versa. If two sets A and B have the same elements, we write A = B. One approach is to list the elements of both sets and verify that they are identical. Another method is to use set notation and set-builder notation to express the elements of each set and then compare them. It is essential to consider both directions, ensuring that no elements are missing or extra in either set, to establish the equality of sets.

2. Can equivalent sets have different subsets?

Yes, equivalent sets can have different subsets. The notion of equivalence between sets is solely based on their cardinality or number of elements. While equivalent sets have the same size, they may contain different elements. As a result, the subsets of equivalent sets can vary because subsets are determined by the specific elements within a set. Even though the overall count of elements remains equal, the specific elements in the subsets may differ.

3. Can equivalent sets have different sizes?

No, equivalent sets cannot have different sizes. Equivalent sets, by definition, have the same cardinality or number of elements. If two sets are equivalent, it means that they contain the same number of elements, even if the elements themselves may differ. So, equivalent sets must have the same size. The concept of equivalence is based on comparing the cardinality or quantity of elements in sets, ensuring that they correspond one-to-one.

4. Are equal sets always equivalent?

Yes, equal sets are always equivalent. When two sets are equal, it means that they have exactly the same elements. Since equivalence is based on the cardinality or number of elements in a set, if two sets are equal, they automatically have the same number of elements. Therefore, equal sets are a special case of equivalent sets where not only do they have the same cardinality, but they also have identical elements. In other words, equality implies equivalence, but equivalence does not necessarily imply equality.

5. Can equal sets have different elements in different order?

No, equal sets cannot have different elements in different order. When two sets are equal, it means that they have exactly the same elements, and the order of the elements does not matter. The equality of sets is not affected by the arrangement or order of the elements within them. Whether the elements are listed in a different order or not, as long as the elements themselves are the same, the sets are still considered equal.