# CHAPTER 1 : Sets

## Outline

1. Introduction
2. Sets and their Represetations
• Set : a set is a well-defined collection of objects.
• Examples of some special sets:
• N : the set of all natural numbers
• Z : the set of all integers
• Q : the set of all rational numbers
• R : the set of real numbers
• Z+ : the set of positive integers
• Q+ : the set of positive rational numbers
• R+ : the set of positive real numbers
• Representation of a set:
• Roster or tabular form is used to list all the elements of a set, separated by commas and enclosed within braces { }.
• Set-builder form is used when all the elements of a set can be defined by a single common property.
3. The Empty Set
• Definition 1 :   A set which does not contain any element is called the empty set or the null set or the void set.
4. Finite and Infinite Sets
• Definition 2 :   A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.
5. Equal Sets
• Definition 3 :   Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A ≠ B.
6. Subsets
• Definition 4 :   A set A is said to be a subset of a set B if every element of A is also an element of B.
AB if whenever aA, then a ∈ B.
AB if a ∈ A => aB.
1. Subsets of set of real numbers
2. Intervals as subsets of R
7. Power Set
• Definition 5 :   The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.
If A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2m.
8. Universal Set
9. Venn Diagrams
10. Operations on Sets
1. Union of sets
• Definition 6 :   The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both).
AB = { x : xA or xB }
• Properties:
1. AB = BA (Commutativity)
2. (AB) ∪ C = A ∪ (BC) (Associativity)
3. A ∪ ∅ = A (Law of identity)
4. AA = A (Idempotent law)
5. UA = U (Law of U)
2. Intersection of sets
• Definition 7 :   The intersection of two sets A and B is the set of all those elements which belong to both A and B.
AB = { x : xA and xB }
• Properties:
1. AB = BA (Commutativity)
2. (AB) ∩ C = A ∩ (BC) (Associativity)
3. ∅ ∩ A = ∅, UA = A; (Law of of ∅ and U)
4. AA = A (Idempotent law)
5. A ∩ (BC) = (AB) ∪ (AC) (Distributive Law: ∩ distributes over ∪)
3. Difference of sets
AB = { x : xA and xB }
11. Complement of a set
• Definition 8 :   Let U be the universal set and A a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A. Symbolically, A′; is used to denote the complement of A with respect to U.
A′ = {x : xU and xA }.
=> A′ = UA.
• Properties:
1. Complement Laws :
2. De Morgan's Law :
3. Law of double complementation :
4. Laws of empty set and universal set :
(i) AA′ = U
(i) (AB)′ = A′B′
(i) (A′)′ = A
(i) ∅′ = U
(ii) AA′ = ∅
(ii) (AB)′ = A′B′

(ii) U′ = ∅
12. Practical Problems on Union and Intersection of Two Sets