A Brief introduction to set and sets types. in very simple method that can every one enjoy during study. #inshallah.

1. Mathematics.
Chapter: Sets
Awais Bakshy

2. Sets.
 Set.
 A set is a collection of well-defined and distinct objects.
 Well-defined collection of distinct objects.
 Well-defined is means a collection which is such that, given any object, we
may be able to decide whether object belongs to the collection or not.
 Distinct objects means objects no two of which are identical (same)
 Set is denoted by capital letters. A, B, C, D, E, etc.
 Elements or members of set.
 The objects in the set are called elements or members of set.
 The elements or members of sets are In small letters.

3. Sets.
 Representation of a set .
 A set can be represented by three methods.
 Descriptive Method.
 A set can be described in words.
 For Example.
 The set of whole numbers.
 Tabular Method.
 A set can be described by listing its elements within brackets.
 For example.
 A={1, 2, 3, 4} B={3, 5, 7, 8,}
 Set Builder Method.
 A set can be described by using symbols and letters.
 For example.
 A={x/x ϵ N }

4. Sets.
 Some Sets.
 Integers.
 Positive Integers + 0 (Zero) + Negative Integers.
 Denoted By Capital Z.
 Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …}
 Negative Integers.
 Denoted by 𝑍′
(Z Dash)
 𝑍′
= {-1,-2, -3, -4, -5, -6, …}
 Natural Numbers.
 N ={1, 2, 3, 4, 5, 6, 7, …}
 Whole Numbers.
 W = {0, 1, 2, 3, 4, 5, 6, 7,8, …}

5. Sets.
 Even Numbers.
 E= {0, 2, 4, 6, 8, 10, 12, 14, …}
 Odd Number.
 O ={1, 3, 5, 7, 9, 11, 13, …}
 Real Numbers.
 R= Q+ 𝑄′
 Irrational Numbers.
 𝑄′
={ 3, 5, 11, 7, … }
 Rational Numbers.
 Q= {
𝑝
𝑞
| p, q ϵ Z }

6. Sets.
 Kinds of sets.
 Finite Set.
 A set in which all the members can be listed is called a finite set.
 A set which has limited members.
 For example.
 A={1, 2,3,4,5,6} B={3,5,7,9} C={0,1,2,3,4,5,6} D={2,4,8,10}
 Infinite Set.
 In some cases it is impossible to list all the members of a set. Such sets are
called infinite sets.
 A set which has unlimited members is called infinite set.
 For example.
 Z= {0, ±1, ±2, ±3, ±4, ±5, ±6, ±7, …}
 W = {0, 1, 2, 3, 4, 5, 6, 7,8, …}
 N ={1, 2, 3, 4, 5, 6, 7, …}
 O ={1, 3, 5, 7, 9, 11, 13, …}

7. Sets.
 Subset.
 When each member of set A is also a member of set B, then A is a subset of B.
 It is denoted by A ⊆ B (A is subset of B)
 For Example.
Let A={1, 2, 3,} B={1, 2, 3, 4,}
A ⊆ B and B ⊇ A
 Types of subset.
 There are two types of Subset.
 Proper subset.
 If A is subset of B and B contains at least one member which is not a member of
A, then A is Said to be proper subset of B.
 It is Denoted by A ⊂ B ( A is proper subset of B).
 For Example.
Let A={1, 2, 3,} B={1, 2, 3, 4,}
A ⊂ B

8. Sets.
 Improper subset.
 If A is subset of B and A=B, then we say that A is an improper subset of B . From
this definition it also follows that every set A is an improper subset of itself.
 For example.
Let A={a, b, c} B={c, a, b} and C={a, b, c, d}
A ⊂ C , B ⊂ C But A=B and B=A
 Equal Set.
 Two sets A and B are equal i.e., A=B “iff” they have the same elements that’s is,
if and only if every element of each set is an element of the other set.
 It is denoted by = (Equal) “iff”(if and only if)
 For Example.
Let. A={1,2,3} B={2,1,3}
A=b

9. Sets.
 Eqiulent set.
 If the elements of two sets A and B can be paired in such a way that each element of A is paired
with one and only one element of B and vice versa, then such a pairing is called a one to one
correspondence “ between A and B”.
 For example.
 If A={Bilal, Yasir, Ashfaq} and B={Fatima, Ummara, Samina} then six different (1-1)
correspondence can be established between A and B. Two of these correspondence are given
below .
i. {Bilal, Yasir, Ashfaq}
{Fatima, Ummara, Samina}
i. ii. {Bilal, Yasir, Ashfaq}
{Fatima, Ummara, Samina}

10. Sets.
 Two sets are said to be Eqiulent if (1-1) correspondence can be established
between them.
 The symbol ~ is used to mean is equivalent to , Thus A~B.
 Singleton Set.
 A set having only one element is called a singleton.
 For example.
A={5} B={8} C={0}
 Empty Set and Null Set.
 A set which contains no elements is called an empty set ( or null set).
 An empty set is a subset of ant set.
 It is denoted by { } , ∅
 For example.
A={ } B={∅} C=∅

11. Sets.
 Power set.
 A set may contain elements, which are sets themselves. For example if: S= set of
classes of a certain school, then elements of C are sets themselves because each
class is a set of students.
 The power set of set S denoted by P(S) is the set containing all the possible
subsets of S.
 For Example.
A={1, 2, 3}, then
P(S)={∅,{1}, {2},{3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} }
 The power set of the empty set is not empty.
 2 𝑚 is formula for finding numbers elements in the power set.
 Universal Set.
 The set which contain all the members/elements in the discussion is known as
the universal set.
 Denoted by U.

12. Sets.
 Union of two sets.
 The union of two sets A and B is denoted by A ∪ B (A union B), is the set of all
elements, which belongs to A or B.
 Symbolically.
A ∪ B={x/x ∈ A ∨ x ∈ B }
 For example.
If A={1,2} and B={7,8} then.
A ∪ B={1,2,7,8}
 Intersection of Two sets.
 The intersection of two set A and B is the of elements which are common to both
A and B.
 It is denoted by A ∩ B (A intersection of B).
 For Example.
If A={1, 2, 3,4, 5, 6} and B={2, 4, 8, 10} then,
A ∩ B={2, 4}

13. Sets.
 Disjoint Set.
 If the intersection of two sets is the empty set then the sets are said to be
disjoint sets.
 For example.
 If A= Set to odd numbers and B= set of even numbers , then A and B are disjoint
sets.
 Overlapping Set.
 If intersection to two sets is non empty set is called overlapping set.
 For example.
If A={2,3,7,8} and B={3,5,7,9}, then
A ∩ B={3,7}
 Complement of a Set.
 Given the universal set U and a set A, then Complement of set A (denoted By 𝐴′
or 𝐴 𝑐
) is a set containing all the elements in U that are not in A.
 Complement of a set = A+𝐴′
= U

14. Sets.
 For example.
 If U={2,3,5,7,11,13} and A={2,3,7,13} , then
𝐴′= U -A={5,7}
 Difference two a set.
 The difference set of two sets A and B denoted by A-B consists of all the
elements which belongs to A but do not belong to B.
 The difference set of two sets B and A denoted by B-A consists of all the
elements which belongs to B but do not belong to A.
 Symbolically.
 A-B={x/x∈ A ∧ x ∉ B } and B-A={x/x∈ B ∧ x ∉ A }

15. Sets.
 For example.
 If A={1,2,3,4,5} and B={4,5,6,7,8,9,10} , then find A-B and B-A.
 Solution.
A-B=?
A-B={1,2,3,4,5}-{4,5,6,7,8,9,10}
A-B={1,2,3}
B-A=?
B-A={4,5,6,7,8,9,10}-{1,2,3,4,5}
B-A={6,7,8,9,10}.
 Note.
 A-B≠B-A.

16. Sets.
 Properties of Union and intersection.
 Commutative property of union.
 Example.
If A={1,2,3,4} and B={3,4,5,6,7,8} then prove A ∪ B =B ∪ A.
Solution.
L.H.S= A ∪ B
A ∪ B={1,2,3,4} ∪ {3,4,5,6,7,8}
A ∪ B={1,2,3,4,5,6,7,8}
R.H.S= B ∪ A
B ∪ A={3,4,5,6,7,8} ∪ {1,2,3,4}
B ∪ A={1,2,3,4,5,6,7,8}
Hence it is proved that A ∪ B =B ∪ A so it holds commutative property of union.

17. Sets.
 Properties of Union and intersection.
 Commutative property of intersection.
 Example.
If A={1,3,4} and B={3,4,5,6,7,} then prove A ∩ B = B ∩ A.
Solution.
L.H.S= A ∩ B
A ∩ B ={1,3,4} ∪ {3,4,5,6,7}
A ∩ B ={3,4,}
R.H.S= B ∩ A
B ∩ A ={3,4,5,6,7} ∪ {1,3,4}
B ∩ A ={3,4}
Hence it is proved that A ∩ B = B ∩ A so it holds commutative property of
intersection.

18. Sets.
 Associative property of union.
 Example.
If A={1,2,3,4} , B={3,4,5,6,7,8} and C={7,8,9} then prove A ∪ ( B ∪ C)=(A ∪ B) ∪ C.
Solution.
L.H.S= A ∪ ( B ∪ C)
B ∪ C=?
B ∪ C ={3,4,5,6,7,8} ∪ {7,8,9}
B ∪ C ={3,4,5,6,7,8,9}
A ∪ ( B ∪ C)={1,2,3,4} ∪ {3,4,5,6,7,8,9}
A ∪ ( B ∪ C) ={1,2,3,4,5,6,7,8.9}
R.H.S=(A ∪ B) ∪ C
A ∪ B=?
A ∪ B={1,2,3,4} ∪ {3,4,5,6,7,8}
A ∪ B={1,2,3,4,5,6,7,8}
(A ∪ B) ∪ C={1,2,3,4,5,6,7,8} ∪{7,8,9}
(A ∪ B) ∪ C={1,2,3,4,5,6,7,8,9}
Hence it is proved that A ∪ ( B ∪ C)=(A ∪ B) ∪ C so it holds Associative property of union.

19. Sets.
 Associative property of intersection.
 Example.
If A={1,2,3,4} , B={3,4,5,6,7,8} and C={4,7,8,9} then prove A ∩( B ∩ C)=(A ∩ B) ∩ C.
Solution.
L.H.S= A ∩ ( B ∩ C)
B ∩ C=?
B ∩ C ={3,4,5,6,7,8} ∩ {4,7,8,9}
B ∩ C ={4,7,8}
A ∩ ( B ∩ C)={1,2,3,4} ∩ {4,7,8}
A ∩ ( B ∩ C) ={4}
R.H.S=(A ∩ B) ∩ C
A ∩ B=?
A ∩ B={1,2,3,4} ∩ {3,4,5,6,7,8}
A ∩ B={3,4}
(A ∩ B) ∩ C={3,4} ∩{4,7,8,9}
(A ∩ B) ∩ C={4}
Hence it is proved that A ∩ ( B ∩ C)=(A ∩ B) ∩ C so it holds Associative property of intersection

20. Sets.
 Distributive property of union over intersection.
 Example.
If A={1,2,3,4} , B={3,4,5,7,8} and C={4,5,7,8,9} then prove A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Solution.
L.H.S= A ∪( B ∩ C)
B ∩ C=?
B ∩ C ={3,4,5,7,8} ∩ {4,5,7,8,9}
B ∩ C ={4,5,7,8}
A ∪ ( B ∩ C)={1,2,3,4} ∪ {4,5,7,8}
A ∪ ( B ∩ C) ={1,2,3,4,5,7,8}
R.H.S = (A ∪ B) ∩ (A ∪ C)
A ∪ B=?
A ∪ B={1,2,3,4} ∪ {3,4,5,7,8}
A ∪ B={1,2,3,4,5,7,8}
A ∪ C=?
A ∪ C={1,2,3,4} ∪ {4,5,7,8,9}
A ∪ C={1,2,3,4,5,7,8,9}
(A ∪ B) ∩ (A ∪ C) = {1,2,3,4,5,7,8} ∩ {1,2,3,4,5,7,8,9}
(A ∪ B) ∩ (A ∪ C) = {1,2,3,4,5,,7,8}
Hence it is proved that A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C) so it holds distributive property of union over intersection .

21. Sets.
 Distributive property of intersection over Union.
 Example.
If A={1,2,3,4} , B={3,4,5,7,8} and C={4,5,7,8,9} then prove A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Solution.
L.H.S= A ∩( B ∪ C)
B ∪ C=?
B ∪ C ={3,4,5,7,8} ∩ {4,5,7,8,9}
B ∪ C ={3,4,5,7,8,9}
A ∩ ( B ∪ C)={1,2,3,4} ∩ {3,4,5,7,8,9}
A ∩ ( B ∪ C)={3,4}
R.H.S = (A ∩ B) ∩ (A ∩ C)
A ∩ B=?
A ∩ B={1,2,3,4} ∩ {3,4,5,7,8}
A ∩ B={3,4}
A ∩ C=?
A ∩ C={1,2,3,4} ∩ {4,5,7,8,9}
A ∩ C={4}
(A ∩ B) ∩ (A ∩ C) = {3,4} ∩{4}
(A ∩ B) ∩ (A ∩ C) = {3,4}
Hence it is proved that A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C) so it holds distributive property of intersection over union .

22. Sets.
 Exercise 1.
 Q N0 1: Prof all conditions of De Morgan's law is A={a,b,c,d,e,f,g,h} and B={a,b,c,e,g,i,j}
 (A ∪ B)′=𝐴′ ∩ 𝐵′
 (A ∩ B)′
=𝐴′
∪ 𝐵′
 Q No 2: Solve all properties of union and intersection If A={2,4,6,8,10} , B={1,2,3,4,5,6}
and C={0,1,2,3,4,5,6,7,8,9,10}
 A ∪ B =B ∪ A.
 A ∩ B = B ∩ A
 A ∪ ( B ∪ C)=(A ∪ B) ∪ C
 A ∩( B ∩ C)=(A ∩ B) ∩ C
 A ∪( B ∩ C) = (A ∪ B) ∩ (A ∪ C).
 A ∩( B ∪ C) = (A ∩ B) ∪ (A ∩ C)