Set Theory: Overview, Questions, Preparation

A well-defined collection of objects is known as a set. The members of a particular set are known as the elements of the set. There is no limit on the number of elements of a set. The elements in a set can be ordered in any manner, and the set remains the same.

Representation of Sets:

A capital alphabet denotes a set’s name while a lower case letter indicates the elements or objects in it. Greek letter epsilon represents ‘belongs to’ in a set.

If M is a set and m1, m2, m3, ... are the members of the set M,

then this can be represented as m1 M.

There are two ways to represent complete sets—the roster form and the set builder form.

• Roster form: 

This tabular form of set representation lists all members of the set in curly braces {}, separated by commas. For example- M= {p, q, r, s, t}.

• Set builder form:

The set builder form uses the common property of the members of a set to represent them symbolically. For example- M= {p : p is a prime number}.

Symbols of Set Theory:

Z: Z represents the set of all integers. Z+ symbolises the set of positive integers and Z- symbolises the set of negative integers.

N: N represents the set of all the natural numbers. 

Q: Q symbolises the set of all rational numbers.

R: R symbolises the set of all real numbers.

C: C symbolises the set of all complex numbers.

Operations on Sets:

Some operations that are performed on sets are-

1. Union of sets
2. Intersection of sets
3. Set difference
4. Symmetric difference
5. Complement of sets
6. Power set
7. Cartesian product of sets

Importance and Weightage

The topic “set theory” is a part of the chapter “Sets” in the syllabus of class 11 maths. This chapter holds a weightage of around 8 marks in the examinations. The knowledge of set theory is crucial to understand “Relations and Functions,” which is a part of the syllabus of class 12 maths.

1. Assume that P (M) = P (N). Show that M = N.

Solution:

To show that M = N

According to the question, P (M) = P (N)

Let x be an element of set M,

x ∈ M.

Since P (M) is the power set of set M, it has all the subsets of set M.

M ∈ P (M) = P (N)

Let K be an element of set N

For any K ∈ P (N),

We have, x ∈ K, K ⊂ N

∴ x ∈ N, ∴ M ⊂ N

Similarly, we have, N ⊂ M

If M ⊂ N and N ⊂ M

∴ M = N.

2. Using properties of sets, show that: M ∪ (M ∩ N) = M.

Solution:

We know that,

M ⊂ M, M ∩ N ⊂ M

∴ M ∪ (M ∩ N) ⊂ M (i)

Also, according to the question,

M⊂ M ∪ (M ∩ N) (ii)

Hence, from equation (i) and (ii)

M ∪ (M ∩ N) = M.

Q: Who introduced the concept of sets?

Q: What is an empty set?

Q: How can we represent positive and negative rational numbers? What does the symbol represent?

Q: How are sets represented with the help of Venn diagrams?

Q: What is the application of sets?