A superset is a set that contains all the elements of another set. For example, if set A = {21, 22, 23, 24} and set B = {21, 23, 24}, we say that set A is the superset of set B. Because all the elements of B [(i.e.,)21, 23, 24] are in set A.
The following illustration shows the relationship between the set and its superset using a Venn Diagram.
Set A is termed the superset of set B (denoted as A ⊇ B) if all elements of set B are also elements of set A. In other words, every element in B must be contained in A.
Superset Symbol
The superset symbol, often known as the superset operator, is a mathematical symbol that represents the notion of one set being a superset of another.
Here's what the symbols ⊃ and ⊇ stand for:
1) ⊃ (Superset Symbol): This symbol represents a strictly superset connection. If A ⊃ B, then set A is a superset of set B and must contain at least one element not found in set B. In other words, A is larger than B and contains items not present in B.
2) ⊇ (Superset or Equal Symbol): This symbol represents a superset connection that may or may not be equal. If A ⊇ B, it signifies that set A is a superset of set B, and it may include all of the same items as set B, as well as more elements.
Example: Let Y = {21, 22, 23, 24, 25, 26} and X = {21, 22, 23, 25, 26}
In the two sets above, every element of X is also an element of Y, and the number of elements of X is smaller than the number of elements of Y.
In other words, n(x) = 4 and n(Y) = 6
⇒ n(x) < n(Y)
As a result, Y is the superset of X.
Other than this example, we can give all the general sets as supersets of each other as follows:
N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C
Where,
- N is the set of Natural Numbers,
- W is the set of Whole Numbers,
- Z is the set of Integers,
- Q is the set of Rational Numbers,
- R is the set of Real Numbers, and
- C is the set of Complex Numbers.
Proper Superset
A proper superset is also called a strict superset. A set B is a proper superset of set A if it contains all the elements of set A and at least one additional element that is not present in set A.
Example, consider the following four sets:
A = 3
B = 4
C = 3
D = 5
From the sets given above:
- B is a proper superset of A because B contains all the elements of A along with one additional element (4).
- C is a superset of A but not a proper superset because both sets contain exactly the same elements (A = C).
- D is not a superset of A because D does not contain the element 3, which is present in A.
Superset vs Subset
| Superset | Subset |
|---|---|
| A set that contains all the elements of another set and possibly more. | A set that contains only a portion of another set. |
| A superset includes one or more other sets. | A subset is included within another set (its superset). |
| Usually has a larger number of elements. | Usually has a smaller number of elements. |
| Symbolized as "⊇" or "⊃" in set theory. | Symbolized as "⊆" or "⊂" in set theory. |
| If A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5, 6}, then B is a superset of A. | If A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, then A is a subset of B. |
Properties of Superset
The following are the main qualities of a superset:
- Every set is a superset of itself.
- Each set is a subset of itself.
- A set has an endless number of supersets.
- Because the null set includes no items, we may claim that any set is a superset of an empty set, for example, every set H would be represented as H ⊃ φ
- Set B is the superset of set A if A is a subset of B.