Subsets: Overview, Questions, Preparation

The concept of set is being used in every branch of mathematics, making it a fundamental part. A set is a well-defined collection of objects. Each element in a set is separated by a comma and enclosed within the curly brackets, often known as braces { }.

Subsets
A subset is a set consisting of the elements which are members of another set.

Consider, 

A = set of all books in your collection. 

B = set of all novels in your collection.

B is a subset of A as every element of B is also an element of A

It is represented by the symbol ‘⊂’, which means ‘is a subset of’.

Super Set: As set B is a subset of set A, A is known as a superset of B and is denoted as A ⊇ B. It is represented using the symbol ‘⊇’ which means ‘is a superset of’.

Proper set: If A and B are two sets, then B is called the proper subset of A if B ⊆ A but A ⊇ B, i.e., B ≠ A. It is represented using the symbol ‘⊂’ which means a proper subset. It is written as A ⊂ B, symbolically. It means that there is at least one element in the set A, which does not belong to set B.

Power set: The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.

Consider,

X = {1, 4, 5, 10, 15}

Y = {1, 5, 15}

Here, Y ⊂ X. 

Whenever a ∈ Y, then a ∈ X. 

In the given consideration, the value of a can be 1 or 5 or 15.

It is represented by the symbol ‘⇒’ which means ‘implies.’ 

Y ⊂ X, if a ∈ Y ⇒ a ∈ X

This statement can be read as, “Y is a subset of X, if a is an element of Y implies that a is also an element of X.”

Points to Remember

Every set is considered to be a subset of itself. 

Empty set denoted as ‘φ’ is a subset of every set as it contains no elements.

No set is a proper subset of itself.

Empty set is a proper subset of every set.

The following related topics are included in the chapter:

1. Empty sets
2. Finite and infinite sets
3. Equal sets
4. Universal sets
5. Venn Diagrams
6. Laws of set theory

Importance and Weightage

This topic covers introductory aspects of a subset in Class 11. The total weightage to the Sets and Functions chapter is 23 marks.

1.Make correct statements by choosing the correct symbols ⊂ or ⊄
Solution:  a{2, 3, 4} ... {1, 2, 3, 4, 5}
                     All elements of set {2, 3, 4} lie in set {1, 2, 3, 4, 5}.
                    Hence, {2, 3, 4} ⊂ {1, 2, 3, 4, 5}
  {a, b, c} ... {b, c, d}
                    Element ‘a’ lies in set {a, b, c} and doesn't lie in set {b, c, d}.
                    Hence {a, b, c} ⊄ {b, c, d}
 {x: x is a triangle in a plane}...{x: x is a rectangle in the plane}
                   Triangle and rectangle are two different diagrams.
                   Hence, {x: x is a triangle in a plane} ⊄ {x: x is a rectangle in the plane}

2. Write all the subsets of the set {1, 2, 3} 
Solution: Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and {1, 2, 3}

3. A = { x, m, p , q}
    B = { x, u , t}
State whether the following statement is true or false:
Solution: B ⊄ A: True

Q: What is a set?

Q: What is a subset?

Q: Which is an empty set?

Q: What are the two kinds of subsets?

Q: What is a power set?