Definition Formula and Solved Examples of Equivalent Sets
Set Theory is a branch of mathematics where we learn different types of sets and their properties. In Maths, sets are defined as a collection of well-defined objects or elements. These objects are also known as elements or members of a set. Sets are represented in two forms i.e set-builder form or roster form. A set is represented by a capital letter. The number of elements in the finite set is called the cardinal number of a set and represented in a curly bracket {…}. Suppose, set A is a collection of all the natural numbers. It is represented as A = {1,2,3,4,5,6,7,8,…..∞}.
In set theory, two sets can either be equivalent, equal or unequal to each other. In this article, we will define equal sets, what is meant by equal and equivalent sets with examples and also the difference between them.
Equivalent Set Definition
Two sets are said to be equivalent if their cardinality number is the same. This means that there must be one to one correspondence between elements of both sets. Here, one to one correspondence means that for each element in set A, there exists an element in set B until sets get exhausted.
If two sets A and B have the same cardinality then there exists an objective function from set A to B.
Two sets A and B are said to be equivalent if they have the same cardinality number i.e. n(A) = n(B).
Generally, we can say that two sets are equivalent to each other if the number of elements in both sets is equal. It is not necessary that both the set have similar elements, or they are a subset of each other.
Equal Set
Equal Set Definition - Two sets A and B are said to be equal only if each element of set A is also present in an element of the set B. In another way, we can say if two sets are the subsets of each other, they are said to be equal. It is represented by:
A = B
A ⊂ B and A ⊂ B ⇔ A = B
If it doesn’t satisfy the above condition, then the sets are said to be unequal. It is represented by:
A ≠ B
Equal Set Example
If A = {1, 3, 8, -2, −7} and B = {-2, −7, 3, 1, 8,}, then A = B. We can note that no matter how many times an element is repeated in the set, it is counted only once. Also, the order of elements doesn’t matter in a set. So, if we represent in terms of cardinal number, we can say that:
If A = B, then n(A) = n(B) and for any x ∈ A and x ∈ B .
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In the above diagram, these are the two sets of different shapes. These are equal sets because the number of elements is the same and their elements are also the same.
Symbol of Equal Set
Equal sets are represented by a symbol of “=” i.e. equality.
Unequal sets are represented by the symbol of “≠” i.e. not equal to.
As in the above example,
A = B i.e. Set A is equal to Set B
But, A ≠ C i.e. Set A is not equal to Set C.
Equivalent Set Example
If P = {1,−7,200, 90,55} and Q = {1,2,3,4,5}, then P is equivalent to Q.
If C = {x : x is positive integer} and D = {d : d is a natural number}, then C is equivalent to D.
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In the above diagram, all the four sets are equivalent sets because the number of elements is the same in all the four sets of triangle, smiley, star and heart. Here n(Triangle) = n(Smiley) = n(Stars) = n(Heart) = 6.
Equivalent Sets Symbol
Equivalent sets are represented by an equivalent sign or “ ~” or “≡”
As in the above example, P ~ Q i.e. Set P is equivalent to Set Q.
Following are the difference between Equal and Equivalent Sets :
Difference Between Equal and Equivalent Sets
Equal Sets and Equivalent Sets
Relation Between Equal and Equivalent Sets
Two sets are equal if they have exactly the same element because their elements and the number of elements both are the same without any order and repetition of elements.
Two sets are equivalent if their number of elements are the same. These elements can be the same or different, but the number should be the same.
Equal sets are equivalent also
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In the above diagram, we can see both the sets are equal because they’re all the elements are the same and their number of elements are also the same so these are equivalent also.
So all equal sets are equivalent also.
Equivalent sets may not be equal sets
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Here X and Y are equivalent sets as a number of elements are the same i.e. 4, but these are not equal sets because elements of both sets are not the same.
So we can say all equivalent sets are not equal sets.
Solved Problems:
Classify the Following Sets as Equivalent or Equal Sets?
(i) A = It contains a set of vowels in the English alphabet.
B = It contains a set of all letters in the word “VOWEL”
(ii) E = The set E={ x : x is a letter in the word “LIFE”}
F = {F, I, L, E}
Sol : Consider set A and B
A = {a, e, i o, u}
B = {V, O, W, E, L}
n(A) = 5, n(B) = 5
Since n(A) = n(B), they are equivalent sets.
(ii) consider set E and F
E = { L, I, F, E }
F = {F, I, L, E}
In both sets, we have the same elements. Hence they are equal sets.
Conclusion
As we have discussed, set is a well-defined collection of objects . As we know all the null sets are equivalent to each other. Consider A and B are two sets such that A = B, then A is equivalent to B. It means two equal sets will always be equivalent but the converse of this statement is not always true. All infinite sets are not equivalent to each other. For example, the set of all real numbers and the set of integers are not equivalent to each other.
FAQs on Understanding Equivalent Sets in Set Theory
1. What are equivalent sets in mathematics?
Equivalent sets are sets that have the same number of elements, even if the elements themselves are different. In other words, two sets are equivalent if their cardinality is equal.
- If Set A has 3 elements and Set B also has 3 elements, then A and B are equivalent.
- Example: A = {1, 2, 3} and B = {a, b, c}. Both have 3 elements, so they are equivalent sets.
2. How do you know if two sets are equivalent?
Two sets are equivalent if they have the same number of elements. To check this:
- Step 1: Count the elements in the first set.
- Step 2: Count the elements in the second set.
- Step 3: Compare the counts (cardinality).
3. What is the difference between equal sets and equivalent sets?
The difference is that equal sets have exactly the same elements, while equivalent sets have the same number of elements.
- Equal sets: A = B if every element of A is in B and vice versa.
- Equivalent sets: A ≈ B if n(A) = n(B).
4. Can you give an example of equivalent sets?
An example of equivalent sets is A = {2, 4, 6} and B = {x, y, z}. Both sets contain 3 elements, so n(A) = n(B) = 3. Since their cardinalities are equal, A and B are equivalent sets, even though the elements are different.
5. What is meant by cardinality of a set?
The cardinality of a set is the number of elements in the set. It is usually written as n(A) for set A.
- If A = {1, 3, 5, 7}, then n(A) = 4.
- Equivalent sets always have equal cardinality.
6. Are two empty sets equivalent?
Yes, two empty sets are equivalent because both have 0 elements. The empty set is denoted by ∅, and its cardinality is n(∅) = 0. Since any two empty sets have the same number of elements, they are equivalent sets.
7. What symbol is used to represent equivalent sets?
Equivalent sets are commonly represented by the symbol ≈. If set A is equivalent to set B, we write A ≈ B. This means n(A) = n(B), indicating both sets have the same number of elements.
8. Do equivalent sets have to contain the same elements?
No, equivalent sets do not need to contain the same elements; they only need to have the same number of elements. For example, A = {1, 2, 3} and B = {a, b, c} are equivalent because n(A) = n(B) = 3, even though the elements are completely different.
9. How are equivalent sets related to one-to-one correspondence?
Two sets are equivalent if a one-to-one correspondence can be established between their elements. This means:
- Each element of Set A pairs with exactly one element of Set B.
- No element is left unpaired.
10. Can infinite sets be equivalent?
Yes, infinite sets can be equivalent if they have the same type of infinity, meaning a one-to-one correspondence exists between them. For example, the set of natural numbers {1, 2, 3, ...} and the set of even numbers {2, 4, 6, ...} are equivalent because each natural number n can be paired with 2n. Thus, both sets have the same infinite cardinality.
