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Difference Between Equal and Equivalent Sets for JEE Main 2024

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Last updated date: 13th Jul 2024
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What is Equal and Equivalent Sets: Introduction

To differentiate between equal and equivalent sets: Equal sets and equivalent sets are concepts used to describe relationships between sets. Equal sets refer to sets that have precisely the same elements, meaning every element in one set is also present in the other set, and vice versa. Two sets are considered equal when they have identical members. On the other hand, equivalent sets pertain to sets that may not have the same elements but have an equal number of elements. The cardinality or size of the sets is the same, even though the individual elements might differ. Equal and equivalent sets are fundamental in set theory and provide a basis for studying set operations and comparisons. Let’s understand them further in detail.


Category:

JEE Main Difference Between

Content-Type:

Text, Images, Videos and PDF

Exam:

JEE Main

Topic Name:

Difference Between Equal and Equivalent Sets

Academic Session:

2024

Medium:

English Medium

Subject:

Mathematics

Available Material:

Chapter-wise Difference Between Topics


What is Equal Sets?

Equal sets are sets that have the exact same elements. Two sets A and B are considered equal if and only if every element of set A is also an element of set B, and every element of set B is also an element of set A. This means that the elements in both sets are identical, with no additional or missing elements in either set. The equality of sets is denoted by the symbol "=" and signifies the complete equivalence of the elements between the sets. Understanding equal sets is essential for comparing and analyzing sets, performing set operations, and solving various mathematical problems involving sets. The characteristics of equal sets are:


  • Same Elements: Equal sets have exactly the same elements. Every element in one set is also present in the other set, and vice versa.

  • Set Equality: The equality symbol "=" is used to represent equal sets. If two sets A and B are equal, we write A = B.

  • Element Equality: For any given element, if it belongs to one set, it must also belong to the other set, and vice versa.

  • Cardinality Equality: Equal sets have the same number of elements. The cardinality or size of the sets is identical.

  • Order of Elements: The order of elements does not affect the equality of sets. The elements can be listed in any order without changing the equality.

  • Subset Equality: If two sets are equal, they are also subsets of each other. This means that every element in one set is contained in the other set.


What is Equivalent Sets?

Equivalent sets are sets that have the same cardinality or number of elements, even though the individual elements might differ. Two sets A and B are considered equivalent if they contain the same number of elements, denoted by |A| = |B|. The elements within equivalent sets may vary, but the overall size or quantity of elements remains the same. For example, a set with three apples and a set with three oranges are considered equivalent sets because they both have three elements. Understanding equivalent sets is crucial for comparing and classifying sets based on their cardinality, as well as for studying concepts like counting, bijections, and equinumerosity in mathematics. The characteristics of equivalent sets are:


  • Same Cardinality: Equivalent sets have the same number of elements or cardinality. The size or quantity of elements in the sets is identical.

  • Element Variation: The individual elements in equivalent sets may differ. They can have different elements, but the overall number of elements remains the same.

  • Cardinality Equality: The equality of cardinalities or sizes of the sets is denoted by |A| = |B|. The number of elements in set A is equal to the number of elements in set B.

  • Set Equinumerosity: Equivalent sets are also referred to as equinumerous sets. They exhibit a one-to-one correspondence between their elements.

  • Bijection Existence: A bijection or one-to-one correspondence can be established between the elements of equivalent sets, mapping each element of one set to a unique element of the other set.

  • Subset and Superset Relations: Equivalent sets are subsets and supersets of each other. They share the same cardinality, so each set is a subset of the other.


Equal and Equivalent Sets Differences

S.No

Category 

Equal Sets

Equivalent Sets

1.

Definition

Sets with the same elements

Sets with the same cardinality

2.

Element Comparison

Elements in both sets are identical

Elements in the sets may differ, but the number of elements is the same

3.

Set Equality

A = B (both sets are exactly the same)

Sets have different elements, but the same size

4.

Cardinality

Sets have the same number of elements

Sets have the same cardinality (number of elements)

5. 

Example

{1, 2, 3} = {1, 2, 3}

{a, b, c} and {x, y, z} have the same number of elements


These are the key differences between equal and equivalent sets. Equal sets have the same elements, while equivalent sets have the same cardinality but may differ in elements. 


Summary 

Equal sets and equivalent sets are fundamental concepts in mathematics used to compare and relate different sets. Equal sets refer to sets that have precisely the same elements. In other words, every element in one set is also present in the other set, and vice versa. On the other hand, equivalent sets are sets that may not have the same elements but have an equal number of elements. This means that the sets have the same cardinality or size, even though the individual elements might differ. Understanding equal sets helps determine if two sets are identical, while equivalent sets allow for comparing sets based on their size or number of elements.

FAQs on Difference Between Equal and Equivalent Sets for JEE Main 2024

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.