Equivalent Sets

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Equivalent Set and its Significance

Sets is an important topic in Mathematics. The concept of equivalent sets in set theory along with all the other major theories of Sets is explained here. The concepts of equal set, equivalent set, equivalent set symbol, equal set definition, etc are clearly explained here, by the highly experienced teachers at Vedantu for the easy understanding of students. To solve the sums of set theory, students must have an idea of what is equivalent set, what is equal set, and several other related concepts. So, now, let us go through the concepts of set theory with relevant examples.


What is Equal Set

The equal set definition means for the sets to be equivalent. To ensure that they are all equal sets equivalent, they need to maintain a similar cardinality. This represents a scenario where the adjacent sets should not be in a one to one correspondence between the elements that are present in both sets. Here, the term one to one correspondence defines that, for every individual element that is present in set A, there exists an element that is present in set B, the set is over or exhausted.


Equivalent Set Definition in Maths

Definition 1: In a case where two sets are provided in a way that A and B have the same cardinal similarity, there is a form of objectives that functions from set A to B.


Definition 2: Two sets A and B are said to be equivalent if they have the same cardinality i.e. n(A) = n(B). Therefore, according to the general terms where we have to define equal set, we can say that the suggested two sets are equivalent to each other, in a condition where the total number of elements in both the sets stand equal with the equivalent set symbol. However, we need to notice that it is not necessary that they have similar elements, or they are a subset of each other.


Equivalent and Equal Sets Examples

Equal Set Example

If P = {1, 3, 9, 5, −7} and Q = {5, −7, 3, 1, 9,}, then P = Q.

In the situation above, it should be noted that no matter the number of times the numbers repeat by themselves in the set, the instance is only counted once. Therefore, we can say that:

In the situation where A = B, n(A) = n(B) and for any x ∈ A, x ∈ B as well.


Equivalent Set Example

In a set where P = {1,−4,2002100,66} and Q = {1,2,3,4}, Here, P is the equivalent set when compared to Q.


Solved Examples

State if the suggested sets are equal or equivalent sets:

(i) {3, 7, 5} and {5, 3, 7}

(ii) {8, 6, 10, 12} and {3, 2, 4, 6}

(iii) {7,  2, 7, 2, 1} and {1, 2, 7}

(iv) {1 4, 9, 16, 25} and {12, 22, 32, 42, 52


Answer:

(i) Equal sets

(ii) Equivalent sets

(iii) Equal sets

(iv) Equal sets


Q2. State if the sets are equal or equivalent:

(i) {x : x is a set with an odd natural number that is less than 8} and {x : x , a letter that is present in ‘girl’}

(ii) {2, 4, 6, 8, 1} and {a, b, d, e, m}

(iii) {5, 5, 2, 4} and {5, 4, 2, 2}

(iv) {x : x , a letter in STRAND} and {x : x a letter in STANDARD}


Answers:

(i) Equivalent sets

(ii) Equivalent sets

(iii) Equal sets

(iv) Equal sets

(v) Equal sets

FAQ (Frequently Asked Questions)

Q1. What are Equal Sets?

Answer: To help you understand the concept of equal sets, let's say there are two sets C and D that can be equal, only in a situation where every element of set C is also the element of the set D. Also, in a condition when the two sets are the provided subsets of each other, the condition is said to be equal. The situation is presented by:


C = D

C ⊂ D and D ⊂ C ⟺ C = D


In the situation where this condition is not met, we can say that these sets are unequal. Now you know what equal sets are.

Q2. What are the Key Points to Summarise in Sets?

Answer: The key pointers that are stated in the chapter of sets, come in handy when you are summarising the theory part of the chapter, a day or two before exams. Listed below are the key takeaways that might come in handy for you while preparation.

  • In the theory, every null set is equal to the equivalent of each other set.

  • In a situation where there are two sets, say C and D that are the two sets that are equal, then C is equivalent to D.