# Sets : Basic Concepts

Any collection of well defined objects is called a set.

- A set may be described by listing all its members and
then putting curly brackets or braces { }. This is called
*roster or tabular form*. - A set may be described as {x|x has property
*p*}. This is called*rule method or set builder form.* - An
*infinite set*has unlimited number of elements. A*finite set*has finite, countable number of elements. An*empty set or null set or void set*has no elements. It is written as { } or . - The number of (different) elements in a set is called its
*cardinal number*. Thus the cardinal number of a null set is zero, whereas cardinal number of an infinite set is not defined. Cardinal number of a singleton set is 1. - Two sets are called
*equal*(written as A B) if they have the same elements. Two finite sets are called*equivalent*if they have the same number of elements. Thus A B if n(A) = n(B). - Two sets are called
*disjoint*if they have no elements in common. - Two sets are called
*overlapping*if they have some elements in common. - A set that contains all the elements under consideration
in a given problem is called
*universal set*. It is written as U or . - Set A is called
*subset*of B if every element of A is also an element of B. We write it as AB (read as "A is a subset of B" or "A is contained in B"). In such a case, we say BA ("B is a*superset*of A" or "B contains A"). - Set A is called a
*proper subset*of set B if every element of A is element of B but there exists at least one element of B which is not an element of A.

## Exercise

- Write the set of seven colours in a rainbow in

(a) roster or tabular form

(b) rule method or set builder form. - Consider the set of all even natural numbers between 12 and 22 (both inclusive). Write it in

(a) roster or tabular form

(b) rule method or set builder form. - Let A = {3, 5, 7, 9, 11}, then write which of the following statements are correct and which are incorrect.

(a) 3A (b) 5, 7 A (c) 8 A

(d) 5 A (e) {3} A (f) {5, 7} A

(g) 5 A (h) {5} A (i) A. - State whether the following statements are true or false. Justify your answer.

(i) = {0}

(ii) The empty set has no subsets.

(iii) Every set has a proper subset

(iv) {0}

(v) The collection of competent school teachers in India is a set. - Classify the following sets as finite set, infinite set or empty set:

(a) The set of all prime numbers

(b) The set of all even prime numbers > 2

(c) The set of even prime numbers

(d) The set of prime numbers less than ten crores. - Let A = {letters of BOMBAY} and B = {letters of MADRAS}

(a) Are these sets disjoint or overlapping?

(b) Are these sets equal?

(c) Are these sets equivalent?

(d) Describe a universal set for this problem.

(e) Is any of these sets subset of the other?

## Answers

**1.**(a) {Violet, Indigo, Blue, Green, Yellow, Orange, Red}

(b) {x|x is a colour in a rainbow}

**2.**(a) {12, 14, 16, 18, 20, 22}

(b) {x|x is an even natural number and 12x22}.

**3.**(a), (b), (c), (h), (i) are correct and (d), (e), (f), (g) are incorrect.

**4.**(i) False, for {0} is not an empty set

(ii)False, for is a subset of

(iii) False, for has no proper subset (iv) True

(v) False, for this collection is not well defined, as a particular teacher considered

competent by one person may not be considered competent by another person.

**5.**(a) Infinite (b) null set (c) finite set with one element

(d) finite set.

**6.**(a) Overlapping, as they have elements M, A in common

(b) No (c) Yes, as both have five distinct elements

(d) {letters of English alphabet} (e) Neither AB nor BA.