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

1. Write the set of seven colours in a rainbow in
   (a) roster or tabular form
   (b) rule method or set builder form.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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