•2 recomendaciones•425 vistas

Descargar para leer sin conexión

Denunciar

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

Seguir

- 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)