# CHAPTER 1 : Sets

## Outline

- Introduction
- 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.
- 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.- 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.- 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.- 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**.**A**⊂**B**if whenever**a**∈**A**, then a ∈**B**.

**A**⊂**B**if**a**∈ A =>**a**∈**B**.- Subsets of set of real numbers
- Intervals as subsets of R
- 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**.- Universal Set
- Venn Diagrams
- Operations on Sets
- 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).**A**∪**B**= {**x**:**x**∈**A**or**x**∈**B**}*Properties:*1.**A**∪**B**=**B**∪**A**(Commutativity)

2. (**A**∪**B**) ∪**C**=**A**∪ (**B**∪**C**) (Associativity)

3.**A**∪ ∅ =**A**(Law of identity)

4.**A**∪**A**=**A**(Idempotent law)

5.**U**∪**A**=**U**(Law of U)- 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**.**A**∩**B**= {**x**:**x**∈**A**and**x**∈**B**}*Properties:*1.**A**∩**B**=**B**∩**A**(Commutativity)

2. (**A**∩**B**) ∩**C**=**A**∩ (**B**∩**C**) (Associativity)

3. ∅ ∩**A**= ∅,**U**∩**A**=**A**; (Law of of ∅ and**U**)

4.**A**∩**A**=**A**(Idempotent law)

5.**A**∩ (**B**∪**C**) = (**A**∩**B**) ∪ (**A**∩**C**) (Distributive Law: ∩ distributes over ∪)- Difference of sets
**A**−**B**= {**x**:**x**∈**A**and**x**∉**B**} - 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**:**x**∈**U**and**x**∉**A**}.

=>**A′**=**U**−**A**.*Properties:*1. Complement Laws :

2. De Morgan's Law :

3. Law of double complementation :

4. Laws of empty set and universal set :(i)**A**∪**A′**=**U**

(i) (**A**∪**B**)′ =**A′**∩**B′**

(i)**(A′)′**=**A**

(i) ∅′ =**U**(ii)**A**∩**A′**= ∅

(ii) (**A**∩**B**)′ =**A′**∪**B′**

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