Difference between Equal and Equivalent Sets with Examples and Properties
Even though equal and equivalent sets sound like there isn’t much difference between them. These two are similar concepts, yes, but there is a minor difference between them that sets them both apart.
But before we divide into equal and equivalent sets, let us understand what cardinality is. Cardinality is the number of elements inside a set. Now this is important because this will help us understand the difference between equal and equivalent sets.
Equal and equivalent sets are terms used to denote some kind of relationship between two sets. You may think of this as some sort of comparison. Like how you would compare apples to oranges but if there is no standard by which we can compare them, then it would be very difficult to establish anything. If we were to compare them by number, then we could say that there are more apples than oranges or vice versa. Or we could say that there are equal numbers of apples and oranges.
The same way, if we were to compare two sets, we could use cardinality as a standard for comparison.
Let us see how it is done.
Define Equal Sets
To understand Equal Set meaning, Equal Set is defined as two sets having the same elements. Two sets A and B can be equal only on the condition that each element of set A is also the element of set B. Also, if two sets happen to be the subsets of each other, then they are stated to be equal sets.
Continuing our above example, if we were to compare one basket of oranges with another basket of oranges, and if the number of oranges is equal in both the baskets, then this is said to be an example for equal sets.
Equal Sets
An equal set can be represented by:
P = Q
P ⊂ Q and Q ⊂ P ⟺ P equals to Q
It is to be noted that if the condition discussed above is not met, then the set is stated to be unequal.
To elaborate, if the two baskets contained an unequal number of oranges or if one basket contained apples and the other contained oranges of the same number, then these cases are said to be examples for unequal sets.
Unequal sets are represented by
P ≠ Q
Define Equivalent Sets
Equivalent sets meaning in Mathematics holds two definitions.
Equivalent Sets Definition 1 - Let's say that two sets A and B have the same cardinality, then, there exists an objective function from set A to B.
Equivalent Sets Definition 2 - Let's say that two sets A and B are stated to be equivalent only if they have the same cardinality, that is, n(A) = n(B).
Thus, to remain or be equivalent, the sets should possess the same cardinality.
In other words, if there is a basket of apples and a basket of oranges, then if they are of the same number, we can call these as an example for equivalent sets.
This condition means that there should be one to one correspondence between the elements belonging to both the sets. In this context, the one to one condition implies that for each element on the set A, there exists an element in the set B, till both the set A and set B gets exhausted.
Therefore, in general, it can be stated that the two sets remain equivalent to each other if only the number of elements in both the sets remain equal. The sets don't need to hold the same elements, or they stay to be a subset of each other.
Equal And Equivalent Sets Examples
Equal Set Example
If we consider numbers to denote the elements of two sets then we can understand equal and equivalent sets in the following manner.
Let’s understand equal sets with an example,
If M= {1, 3, 9, 5, −7} and N = {5, −7, 3, 1, 9,}, then it can be stated that M = N. It is to be noted that no matter how many times an element is repeated in a particular set, the element is counted only once. Also, it is to be pointed out that the order does not matter for the elements for a specific set. Therefore, in terms of cardinal number, equal sets can be stated that:
If P = Q, then n(P) = n(Q) and for any x ∈ P, x ∈ Q too.
Equivalent Set Example
If S = {x: x, where x is stated to be a positive integer} and T = {d : d, where x is said to be a natural number}, then S is stated to be equivalent to T.
Thus, it can be stated that an equivalent set is simply a set with an equal number of elements. However, the sets don't need to have the same elements but must comprise the same number of elements.
Let’s Understand Equivalent Sets With Examples
If A = {1,−7,200011000,55} and B = {1,2,3,4}, then A is equivalent to B.
If Set G: {Sweater, Mittens, Scarf, Jacket} and Set H: {Apples, Bananas, Peaches, Grapes}, it can be noted that both Set G and Set H comprise word elements in different categories and have the same number of elements i.e. four.
We are now clear on what equal and equivalent sets are. Now let us expand our knowledge to accommodate a few fascinating facts about the relation between equal and equivalent sets. They are mentioned as important pointers below.
Important Points to Remember on Equivalent Sets
All the null sets are said to be equivalent to each other.
Not all the infinite sets remain equivalent to each other. For example, the equivalent set of all the real numbers and the equivalent set of the integers.
If P and Q are stated to be two sets such that P is equal to Q, that is, (P = Q). This example means that two equal sets will always remain to be equivalent, but the converse of the equivalent set may or may not remain true.
An equal set can be an equivalent set, but it is not necessary for an equivalent set to be an equal set.
FAQs on Equal and Equivalent Sets in Set Theory
1. What are equal sets in mathematics?
Two sets are called equal sets if they contain exactly the same elements. In equal sets, every element of one set is also an element of the other set, and order does not matter.
- If A = {1, 2, 3} and B = {3, 2, 1}, then A and B are equal sets.
- Equal sets are written as A = B.
- Repeated elements are ignored because sets do not count duplicates.
2. What are equivalent sets?
Two sets are called equivalent sets if they have the same number of elements, even if the elements are different. Equivalent sets have equal cardinality (size).
- If A = {1, 2, 3} and B = {a, b, c}, then A and B are equivalent.
- The number of elements is called the cardinality of a set.
- Equivalent sets are written as n(A) = n(B).
3. What is the difference between equal sets and equivalent sets?
The main difference is that equal sets have the same elements, while equivalent sets have the same number of elements. Equal sets must match element by element, but equivalent sets only need equal size.
- Equal sets: A = {1,2,3}, B = {1,2,3}
- Equivalent sets: A = {1,2,3}, B = {a,b,c}
- All equal sets are equivalent, but not all equivalent sets are equal.
4. How do you check if two sets are equal?
To check if two sets are equal, verify that every element of one set appears in the other set and vice versa. Follow these steps:
- Step 1: List elements of both sets clearly.
- Step 2: Compare each element.
- Step 3: Ensure no extra or missing elements.
5. How do you determine if two sets are equivalent?
Two sets are equivalent if they have the same number of elements, meaning their cardinalities are equal. To determine this:
- Step 1: Count elements in Set A → n(A).
- Step 2: Count elements in Set B → n(B).
- Step 3: Compare the counts.
6. Can two sets be equal but not equivalent?
No, two sets cannot be equal without being equivalent because equal sets always have the same number of elements. If A = B, then automatically n(A) = n(B). Therefore, all equal sets are also equivalent sets.
7. Can two sets be equivalent but not equal?
Yes, two sets can be equivalent but not equal if they have the same number of elements but different elements. For example:
- A = {1, 2, 3}
- B = {4, 5, 6}
8. What is the cardinality of a set?
The cardinality of a set is the total number of distinct elements in that set. It is denoted by n(A) for a set A.
- If A = {2, 4, 6, 8}, then n(A) = 4.
- Cardinality is used to check equivalent sets.
- Duplicate elements are counted only once.
9. Are two empty sets equal and equivalent?
Yes, all empty sets are both equal and equivalent because they contain no elements. The empty set is denoted by ∅ or {}.
- n(∅) = 0
- Any empty set has zero elements.
- Therefore, all empty sets are equal to each other.
10. What are some common mistakes when dealing with equal and equivalent sets?
A common mistake is confusing equal sets with equivalent sets. Students often think sets with the same number of elements are automatically equal, which is incorrect.
- Equal sets require identical elements.
- Equivalent sets only require equal cardinality.
- Ignoring duplicate elements in sets can also cause errors.
