Superset
1.0What is a Superset?
A superset is a set that includes all the elements of a given set, along with potentially other elements. The relationship between a set and its superset is defined by containment. For example, if we have a set B = {1, 2}and a set A = {1, 2, 3, 4}, then A is a superset of B because all elements of B are also present in A.
This concept is the reverse of a subset. If B is a subset of A, then A is a superset of B. This dual relationship is crucial for understanding set theory.
2.0Definition of a Superset
Let A and B be two sets. We say that A is a superset of B if and only if every element of B is also an element of A.
This can be expressed using mathematical notation: A⊇B⟺∀x(x∈B⇒x∈A)
This means that the statement "A is a superset of B" is logically equivalent to "A contains all elements of B".
3.0Symbol and Notation
The symbol used to denote a superset is ⊇ or ⊃.
- A⊇B means "A is a superset of B." This includes the possibility that A and B are equal.
- A⊃B means "A is a proper superset of B," which implies that A and B are not equal.
The direction of the symbol is important. It always opens towards the larger, or superset, side. A helpful way to remember this is that it's like an alligator's mouth, which always wants to eat the bigger quantity.
4.0Superset vs. Subset: The Key Difference
The relationship between a superset and a subset is a mirror image.
In the given example, A is a subset of B, and conversely, B is a superset of A. The two concepts are inherently linked and describe the same relationship from different perspectives.
5.0Proper Superset
A proper superset is a more specific type of superset. A set A is a proper superset of set B if:
- A is a superset of B (A⊇B).
- A is not equal to B (A=B).
This means that A contains all the elements of B and at least one element that is not in B.
Notation: The symbol for a proper superset is ⊃. So, A⊃B means that A is a proper superset of B.
Example:
- Let A={a,b,c,d} and B={a,b,c}.
- A is a superset of B because every element in B is also in A.
- A is a proper superset of B because A contains the element 'd' which is not in B, and A and B are not equal.
Non-Example:
- Let P={1,2,3} and Q={1,2,3}.
- P is a superset of Q (P⊇Q).
- However, P is not a proper superset of Q because P=Q.
Properties of Proper Superset
- The empty set ϕ is a proper superset of no set. It is a subset of every set, but can't contain elements not in the other set, thus it can't be a proper superset.
- Every set is a proper superset of the empty set ϕ.
- If A is a proper superset of B, then the number of elements in A is strictly greater than the number of elements in B (i.e., ∣A∣>∣B∣).
6.0Solved Examples on Supersets
Understanding supersets is crucial for solving problems in set theory, probability, and relations. Here are some examples typical of the JEE Main and Advanced level.
Example 1: Basic Identification
Question: Let P = {x ∈ N∣ 1 ≤ x ≤ 5} and Q = {x ∈ N∣ 1 ≤ x ≤ 3}. Is P a superset of Q?
Solution:
First, list the elements of each set.
- P={1,2,3,4,5}
- Q={1,2,3}
- To check if P is a superset of Q, we must verify if every element of Q is also in P.
- The elements of Q are 1, 2, and 3.
- These elements are all present in P.
- Therefore, P is a superset of Q. Since P=Q, P is also a proper superset of Q.
Example 2: Union and Intersection
Question: Let A={1,2,3}, B={2,3,4}, and C=A∪B. Is C a superset of both A and B?
Solution:
First, find the union of A and B.
- C=A∪B={1,2,3}∪{2,3,4}={1,2,3,4}.
Now, we check the superset relationship for both sets.
Is C a superset of A?
- The elements of A are 1, 2, 3. All of these are in C. So, C⊇A.
- Is C a superset of B? The elements of B are 2, 3, 4. All of these are in C. So, C⊇B.
Thus, C is a superset of both A and B.
Example 3: Cardinality and Power Set
Question: If a set A has 5 elements, how many of its subsets are proper supersets of a given subset B with 3 elements?
Solution:
This question is framed to be tricky. A proper superset must contain all elements of the original set AND at least one more. However, the question asks for a subset of A that is a proper superset of a subset B. This is a logical contradiction.
Let's break it down:
- We need to find a set C such that C⊆A and C⊃B.
- This means C must contain all elements of B and at least one element from A that is not in B.
- ∣A∣=5 and ∣B∣=3.
- The number of elements in A but not in B is ∣A∣−∣B∣=5−3=2.
- Let the elements of A not in B be {x,y}.
- A set C which is a proper superset of B must contain all elements of B and at least one of {x,y}.
- The possible sets C are formed by taking the set B and adding a non-empty subset of {x,y}.
- The non-empty subsets of {x,y} are: {x}, {y}, {x,y}.
- So, the possible sets C are:
- There are 3 such subsets of A that are proper supersets of B.
7.0Properties of Supersets
- Reflexive Property: Every set is a superset of itself. A⊇A.
- This is because every element of A is also an element of A.
- Transitive Property: If A is a superset of B and B is a superset of C, then A is also a superset of C.
- A⊇B and B⊇C⟹A⊇C.
- This is because if every element of C is in B, and every element of B is in A, then it logically follows that every element of C is in A.
- The Universal Set: The universal set, denoted by U, is a superset of all sets under consideration.
- The Empty Set: The empty set, ϕ, is a subset of every set. This means that every set is a superset of the empty set.
- Relationship with Union and Intersection
- The union of two sets, A∪B, is a superset of both A and B.
- The intersection of two sets, A∩B, is a subset of both A and B. This means both A and B are supersets of their intersection.
- A⊇(A∩B) and B⊇(A∩B).