Set (Definition, Theory, Application)

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  • Topic: Set, Subset, Set theory
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  • Published : December 5, 2012
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Chapter – 1

Set (Definition)
1. Set 2. Sub set 3. Proper Subset 4. Venn diagram 5. Intersection of Sets 6. Union of Sets 7. Universal set / Universe 8. Empty Set or Null Set or Void Set 9. Singleton Set or Unit set 10. Complement of a set or Absolute Complement of a set 11. Difference of two sets 12. Symmetric difference of two sets 13. Power Set 14. Disjoin Set 15. Finite & Infinite set 16. Equal sets 17. Equivalent set 18. Set of Sets or Family of Sets or Class of Sets 19. Ordered Pair 20. Product Set or Cartesian Product or Direct Product 21. Partition of a set

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Composed by Abir Khan th BBA 16 Batch at SMUCT, ID No. 102-401-019 Cell: +880 1678 AHKHAN (245426) Mail: mail@abirkhan.co.cc, Web: www.abirkhan.co.cc

1) Set: A set is a collection of well define & well distinguished objects. The objects are called the elements of a set. A set usually be denoted by capital letter. The elements may be anything such that numbers, persons, river etc. The elements are separated by comma (,) & enclosed in braches [{ }]. Example: i) Roster Method form: ii) Rule Method form: A= {1, 2, 3, 4} B = {x : x is vowel} A A

12 34

U =A

U

A= {1, 2, 3, 4}

2) Subset: The set B is said to be a subset of the set A if every element of B is also an element of A. It is denoted by B  A. Let, A = {1, 2, 3, 4}  B  A  B is the subset of A. B = {1, 2, 3} or A  B  A is the superset of B. Sign of subset =  ; Sign of superset =   How many subsets will be when a set has element? Formula of subset = Let, A = {1, 2, 3}  Number of subset = [2= Fixed, n = No. of element in the set] = 23 =8 Subset of A are, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}  Proper subset {1, 2, 3}, {}  Improper subset Example: Let, B = {1, 2, 3, 4, 5} Venn diagrammed A = {1, 2, 3} Here, A is the subset of B U B A Written as A  B AB  Methods of Set: a) Roster Method b) Rule Method

a) Roster Method: In which method the elements of the set are separated by the comma (,) and enclosed by braches [{ }] is called Roster Method. Or Under this method we just make a list of the objects of the set and put it within braches [{ }]. Example: Let, A = {1, 2, 3, 4} b) Rule Method: This method consists in the listing of property or properties satisfied by the elements of the set. We write {x : x satisfy Properties P} i.e., the set of all those elements such that each element x satisfies the properties P/ Example: 1) Let, A = {x : x is odd} 2) Let, B = {x : x is even, x < 12}

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Composed by Abir Khan th BBA 16 Batch at SMUCT, ID No. 102-401-019 Cell: +880 1678 AHKHAN (245426) Mail: mail@abirkhan.co.cc, Web: www.abirkhan.co.cc

3) Proper Subset: Let A and B be the two sets. Then the set A is called proper subset of B, if every element of A is in B but at least one element of B is not an element of A. It is denoted by A  B Example: Let, B = {1, 2, 3, 4} Venn diagrammed A = {1, 2, 3} Here, A is the subset of B U B A Written as A  B AB 4) Venn Diagram: The Venn diagram is named after English Logician John Venn (1834 – 1923) to present pictorial presentation. The universal set U (say) is denoted by a region enclosed by a rectangle and one or more sets say A, B, C or shown through closed curves within rectangle. U (Rectangle)

A (Closed Circle)

5) Intersection of Sets: Let A and B be the two sets. The intersection of A and B is the set consisting of a element which belong to both A and B. It is denoted by A  B Example: ii) Let A = {1, 2, 3} iii) Let A = {1, 2, 3} i) A  B = {x : x B} B = {2, 3, 4, 5} B = {4, 5, 6}  A  B = {2, 3} A  B = {1, 2, 3}  {4, 5, 6} = { } or Venn diagram of A  B: A B U A 1

2 3

B
4 5

U

A

B

U

6) Union of Sets: Let A and B be the two sets. Then the union of A and B is the set consisting of All elements which belong to either A or B. It is denoted by ‘A  B’ Example: Let, B = {1, 2, 3} Venn diagrammed of A  B A = {3, 4, 5} U A B  A  B = {1, 2, 3, 4, 5,}

 Union of Sets: (i) Let, x AB =x A or x B ...
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