Concept of set.

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Concept of set.

  1. 1. CONCEPT OF SETS.
  2. 2. 1.Way of listing the elements ofSets Put the elements in curly brackets.. {1,2,3,4}
  3. 3. 2.Specifying properties of sets i) intensional definition A is the set whose members are the first four days in a week. ii) extensional definition A = {Sunday, Monday, Tuesday, Wednesday}
  4. 4. 3.Set membership , ∈ ∈ means element of a.k.a relation x ∈ A means x is an element of set A A contains x A x
  5. 5. 4.Empty set.. Empty set is a set having no elements.
  6. 6. 5.Set of numbers (Z,N,etc....)
  7. 7. 6.Set EqualityDefinition: Two sets are equal if and only if they have the same elements.Example:{1,2,3} = {3,1,2} = {1,2,1,3,2}Note: Duplicates dont contribute anything new to aset, so remove them. The order of the elements in aset doesnt contribute anything new. Example: Are {1,2,3,4} and {1,2,2,4} equal?No!
  8. 8. 8.Subset Definition: A set A is said to be a subset of B if and only if every element of A is also element of B. We use A ⊆ B to indicate A is a subset of B. Example: A={1,2,3} B ={1,2,3,4,5} Is: A ⊆ B ? Yes.
  9. 9. POWER SET Power set of S is the set of all subsets of the set S. The power of set S is denoted by P(S). Example: What is the power set of set {3, 4, 5} ?Solution:P({3, 4, 5}) is the set of all subsets of {3, 4, 5}P({3, 4, 5}) = {Ø, {3}, {4}, {5}, {3, 4}, {3, 5}, {4, 5}, {3, 4, 5}.
  10. 10. SET OPERATIONUnionA∪BBoth circle are shaded.
  11. 11. SET OPERATIONIntersectionA∩BOnly the portion shared by both circles are shaded.
  12. 12. OPERATION SET Disjoint setIf the intersection is empty, we called disjoint set. U A B
  13. 13. OPERATION SETSet differenceA-B
  14. 14. OPERATION SETExample set difference :K = {a, b} L = {c, d} M = {b, d}K –L ={a, b}K –M ={a}L – M= {c}K–K=ØK–Ø=KØ–K=Ø
  15. 15. OPERATION SETSet complimentaryEverything in universe that is not in the set
  16. 16. OPERATION SET Example : U={A,B,C,D,E,F,G} C={A,B,D,E,F} C´={C,G} U-C=C´
  17. 17. OPERATION SETCharacteristics of set.
  18. 18. GENERELISE UNION ANDINTERSECTION
  19. 19. GENERELISE UNION ANDINTERSECTIONExample :Let A ={0,2,4,6,8}, B ={0,1,2,3,4} and C ={ 0,3,4,9}.A∪B ∪C :{0, 1 ,2, 3, 4, 6, 7, 8, 9}A ∩ B∩ C :{0}
  20. 20. CARTESIAN PRODUCT A and B are set A and B is set of all ordered pairs(a,b) where A∈a and b∈BExample:Suppose A={1,2,} and B={2,3}. ThenA×B={(1,2),(1,3),(2,2),(2,3) }andB×A={(2,1),(2,2),(3,1),(3,2)}

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