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Sets

• 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)
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