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: $A\supseteq B⟺\forall x(x\in B\Rightarrow x\in A)$

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:

1. A is a superset of B (A⊇B).
2. 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:
• • B∪{x}
  • B∪{y}
  • B∪{x,y}
• There are 3 such subsets of A that are proper supersets of B.

7.0Properties of Supersets

1. Reflexive Property: Every set is a superset of itself. A⊇A.

• This is because every element of A is also an element of A.

2. 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.

3. The Universal Set: The universal set, denoted by U, is a superset of all sets under consideration.

• For any set A, U⊇A.

4. 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.

• For any set A, A⊇ϕ.

5. 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).