What Are Subsets and Supersets? Key Differences and Examples
What is a Subset?
A set refers to an organized collection of objects. In mathematics, the set is any well-defined collection of mathematical objects. Those objects can be anything from the likes, dislikes, ages of people, simple integers, to complex scientific data. It can be a result of a simple coin toss or a dice roll, or it can be an outcome of such experiments repeated hundreds/thousands of times. Roaster form and set builder are the two ways to represent a set. Elements of the set are nothing but the objects inside that particular set. In this article, you can learn about subset and superset meaning and similar concepts. We will also know what is a proper subset and improper subset.
Basics of Sets and Subsets
As you already know, a set is an organized collection of objects grouped within {}. You can further divide them into tiny sets of its own, called as subsets. In math, a set P becomes a subset of the set Q, provided that every element of set P is also an element of set Q. If ‘x’ represents any element of set P, then you can represent it symbolically as:
If x ∈ P and x ∈ Q, then P ⊂ Q (‘⊂’ denotes ‘subset of’) and it reads, P is a subset of Q. Refer fig. 1 as below.
Also, note that the converse is also true: if P ⊂ Q and x ∈ P, then x ∈ Q holds too.
In case the P is not a subset of Q, then you can represent it as P ⊄ Q. You should also know that when P ⊂ Q, it doesn’t mean that all elements of Q are also the elements of P. However if that happens, you can represent it as, P ⊂ Q and Q ⊂ P. It also means that P = Q.
Symbolically, P ⊂ Q and Q ⊂ P ⟺ P = Q. Here, ‘⟺’ stands for ‘if and only if’ (iff).
The above condition leads to fantastic insight. As P = Q, it shows that any set is the subset of itself. As you already know that null or empty set, denoted by ϕ, doesn’t contain any elements. By referring to the above discussion, you can conclude that a null set must be a subset of itself. As it doesn’t have any element, it becomes a subset of every other non-empty set. It leads us to a conclusion that each non-empty set has at least two subsets; the empty set and itself.
What is a Superset?
The conditions that define the superset are, P ⊂ Q and P ≠ Q. When those conditions get fulfilled, you can say that Q is the superset of P. Superset gets denoted with a symbol which is a mirror image of the ‘⊂,’ that gets used to represent a subset.
There are two properties of a superset. First, every set is a superset of a null or empty set. It means P ⊃ ϕ because ϕ has no elements at all. Secondly, now that you know each set is a subset of itself, you can also say that each set is also a superset of itself.
So, if Q is a super of P, then you can represent it as Q ⊃ P. Below are some examples to make the concepts even more clear.
P = {set of all polygons} and Q = {set of regular polygons}. Here, Q ⊂ P and Q ≠ A, hence P is the superset of Q.
A = {1, 2, 3, 4, 5, 6} and B = {m: m<4 and m ϵ N}. Here also, set B is the subset of set A and in contrast, set A is the superset of set B.
Solved Examples
Question 1: If set P = {Mother, sister, brother, you, father} and set Q = {You}, then how is Q ⊂ P?
Answer: Here, set P denotes your family members and set Q holds a single element, which is you. The definition of subset says each element of a subset is part of the original set. From the given information, the element ‘you’ is a part of your family, which is the set P. Therefore, Q is a subset of P, represented as Q ⊂ P.
Question 2: If P = {a: a is an even natural number} and Q = {b: b is a natural number}, then figure out the subset here.
Answer: As per given data, P = {2, 4, 6, 8, 10, 20, . . .} and Q = {1, 2, 3, 4, 5, 6, 7, 8, 9,. . .18, 19, 20, . . .}. As you can see, set Q includes all the elements of set P. So, P is the subset of Q or P ⊂ Q.
FAQs on Sets, Subsets, and Supersets: Concepts Made Simple
1. What is a set in mathematics, and how is it represented?
A set is a well-defined collection of distinct objects or elements. The objects within a set are called its elements or members. Sets are typically represented in two ways:
- Roster Form: Listing all the elements, separated by commas, within braces { }. For example, the set of the first three odd natural numbers is {1, 3, 5}.
- Set-Builder Form: Defining the elements by stating a property they all share. For example, {x | x is a vowel in the English alphabet}.
2. What are subsets and supersets? Explain with an example.
A set 'A' is a subset of another set 'B' if every element of 'A' is also an element of 'B'. This is denoted as A ⊆ B. Conversely, set 'B' is called the superset of set 'A'. For example, if A = {1, 2} and B = {1, 2, 3}, then A is a subset of B because both 1 and 2 are present in B. Consequently, B is the superset of A.
3. What is the main difference between a subset and a proper subset?
The main difference lies in the possibility of the two sets being equal.
- A subset (A ⊆ B) means all elements of A are in B. In this case, set A can be equal to set B.
- A proper subset (A ⊂ B) means all elements of A are in B, but A cannot be equal to B. This implies that B must contain at least one element that is not in A.
4. How do the symbols ⊂ and ⊆ differ in set theory? When should each be used?
These symbols define the precise relationship between two sets. The symbol ⊆ denotes a subset, indicating the first set can be smaller than or equal to the second. The symbol ⊂ denotes a proper subset, strictly meaning the first set must be smaller than the second. You should use:
- ⊆ (Subset): When stating that Set A is a subset of Set B, allowing for the possibility that A and B are equal. For example, {1, 2} ⊆ {1, 2}.
- ⊂ (Proper Subset): When stating that Set A is a subset of Set B and is definitively not equal to B. For example, {1, 2} ⊂ {1, 2, 3}.
5. What is a power set, and how is it related to the number of subsets a set can have?
The power set of a set 'A', denoted as P(A), is the collection or set of all possible subsets of A. The relationship is direct: the number of elements in the power set is equal to the total number of subsets. If a finite set 'A' has 'n' elements, then the number of subsets it has is 2ⁿ. Therefore, its power set, P(A), will contain 2ⁿ elements. For instance, if A = {a, b}, it has n=2 elements, so it has 2² = 4 subsets: ∅, {a}, {b}, and {a, b}. The power set is P(A) = {∅, {a}, {b}, {a, b}}.
6. Why is the empty set (∅) considered a subset of every set?
The empty set, or null set (∅), is considered a subset of every set because it does not violate the fundamental definition of a subset. A set 'A' is a subset of 'B' if there are no elements in 'A' that are not in 'B'. Since the empty set has no elements at all, it is impossible for it to contain an element that is not present in another set. This condition holds true for any set 'B', making the empty set a universal subset.
7. How are sets, subsets, and supersets used to represent relationships between groups of real numbers?
Sets, subsets, and supersets are essential for classifying number systems and showing their relationships. For example:
- The set of Natural Numbers (N) is a proper subset of the set of Whole Numbers (W), so N ⊂ W.
- The set of Whole Numbers (W) is a proper subset of the set of Integers (Z), so W ⊂ Z.
- The set of Integers (Z) is a proper subset of the set of Rational Numbers (Q), so Z ⊂ Q.
- The set of Rational Numbers (Q) is a proper subset of the set of Real Numbers (R), so Q ⊂ R.
