Superset is one of the not-so-common topics in set theory, as this is not used as much as its related term, i.e., Subset. A superset is a set that contains all of the items of another set, known as the subset. We know that if B is contained within A, that means A contains B. In other words, if B is a subset of A, then A is its superset.
In this article, the concept of a superset is discussed in plenty of detail. Other than that, its definition, symbols, properties, and several solved examples as well.
The following illustration shows the relationship between the set and its superset using a Venn Diagram.
If we have two sets, a superset in Maths is a set that includes almost all of the items of the smaller set.
If P and Q are two sets, and P is the superset of Q, and Q is the smaller set, then all of Q's elements are present in P. Components/members of a given set are entities or items that belong to a certain sort of set. In arithmetic, relations, and functions, a set is commonly represented by capital letters, whereas the components are represented by lowercase letters. All of the items are enclosed by braces'{}'.
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Mathematical Definition of Superset
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.
For example,
If set A = {21, 22, 23, 24} and set B = {21, 23, 24} we may say that set A is the superset of B. Because all the elements of B [(i.e.,)21, 23, 24] are in set A.
Superset Symbol
The superset symbol, often known as or, is a mathematical symbol that represents the notion of one set being a superset of another. A set A is said to be a superset of another set B if every constituent of set B is also an element of set A. In other words, set A contains all of the components in set B.
Here's what the symbols ⊃ and ⊇ stand for:
- ⊃ (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.
- ⊇ (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.
These connections can be expressed mathematically as follows:
- A ⊃ B => This can be translated as "A is a strict superset of B" or "A contains all of B's elements and possibly more."
- A ⊇ B => This is equivalent to saying "A is a superset of B, including the possibility that they are equal."
The superset symbol is widely used in mathematics and set theory to express relationships between sets, and it is essential for understanding set inclusion and confinement.
Superset 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 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.
Learn more about the Subset of Real Numbers.
Proper and Improper Superset
A correct superset is often referred to as a stringent superset. If set X is the correct superset of set W, then all of set W's elements are in X, but set X must have at least one member that is not in set W.
Take, for example, four sets.
- W = {u, v, w}
- X = {u, v, w, x}
- Y = (u, v, w}
- Z = {u, v, y}
Because X is not equal to W, it is the proper superset of W from the sets described above.
- Y is a superset of W, however, it is not a proper superset of W because of Y= W.
- Z is not a superset of W since it lacks the element "w" that is present in set W.
Differences Between Superset and Subset
The primary distinction between a superset and a subset is that they are diametrically opposed. Take, for example, two sets, M and N. M = {13, 15, 19} and N = {15, 19}. Then, {13, 15, 19} is the superset of {15, 19}. To put it another way, if M is the superset of N, then N is the subset of M. The distinctions between the superset and subset are listed below.
| Parameter | Superset | Subset |
|---|
| Definition | A set that contains all the elements of another set and possibly more. | A set that contains only a portion of another set. |
|---|
| Relationship | A superset includes one or more other sets. | A subset is included within another set (its superset). |
|---|
| Cardinality | Usually has a larger number of elements. | Usually has a smaller number of elements. |
|---|
| Notation | Symbolized as "⊇" or "⊃" in set theory. | Symbolized as "⊆" or "⊂" in set theory. |
|---|
| Example | 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 it is offered as a subset of set A.
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Solved Examples on Superset
Example 1: Assume M = {m, n, q, r, s} and N = {m, o, p, q}. Is M a subset of N or a superset? Provide reasons as well.
Solution:
Given: M ={m, n, q, r, s} and N= {m, o, p, q}.
Because the elements "o and p" in set N are not present in set M, the supplied set M is not a superset of set N. As a result, we can plainly state that M is not a superset of N. As a result, N does not constitute a subset of M.
Example 2: Determine who is a subset here if M = {x: x is an odd natural number} and N = {y: y is a natural number}.
Solution:
Given: M = {1, 3, 5, 7, 9, 11,13, …} and N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …}.
It is obvious that set N contains ALL of the items of set M. As a result, set M is a subset of set N, or M ⊂ N.
Example 3: Show that M is the correct superset of N if M = {32, 33, 37, 39} and N= {32, 37, 39}. Justify your response.
Solution:
Given:
- Set M = {32, 33, 37, 39}
- Set N = {32, 37, 39}
M is the appropriate superset of set N since all of set N's components are also present in set M, however, we can see that set M contains one more additional element (i.e., 33) than set N.
We can also show that set M is not equivalent to set N.
As a result, we may argue that set M is the correct superset of set N.
Example 4: There are three sets: a set of all real numbers (R), a set of natural numbers (N), and a set of all whole numbers (W). Determine the superset of them.
Solution:
Real numbers include natural numbers, whole numbers, integers, rational numbers, and irrational numbers, but not complex numbers.
Natural numbers include all positive counting numbers indicated by N = {1, 2, 3,...}
Whole numbers are made up of natural numbers plus 0. These are represented by the symbol W = {0, 1, 2, 3,...}
We can now conclude:
- Real numbers ⊃ Natural numbers
- Real numbers ⊃ Whole numbers
Also, Whole numbers ⊃ Natural numbers
Example 5: Check whether the following statements are true or false.
a) An empty set is a superset of every other set.
b) Every set is a superset of the empty set.
c) Every set is a superset of itself.
b) Every set has a limited number of supersets.
Solution:
a) False, as empty set do not contain any element.
b) True , as empty set is subset of all sets.
c) True, as set itself contains all the elements it have.
d) False, as we can add one element to set to make it superset, and that element can be anything.
Practice Problems on Supersets
Problem 1: Let A = {1, 2, 3, 4, 5} and B = {2, 4}. Determine whether each statement is true or false:
a) A is a superset of B.
b) B is a subset of A.
c) B is a proper subset of A.
d) A and B are disjoint sets.
Problem 2: Given the sets C = {red, green, blue} and D = {red, green, blue, yellow}. Is D superset of C or C superset of D?
Problem 3: Consider the sets E = {a, b, c, d}, F = {c, d, e}, and G = {a, e}. Draw a Venn diagram to represent these sets and their relationships. Identify any supersets or subsets.
Problem 4: Let H = {x, y}. Find the power set of H and identify which subset of H are supersets of the set {x}.
Problem 5: Let M be the set of all mammals and C be the set of all carnivores. Determine whether each statement is true or false:
a) M is a superset of C.
b) C is a subset of M.
c) M is a proper superset of C.
d) C is a proper subset of M.
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