Operations on Sets and Venn Diagrams
     
    Operations on sets and Venn diagrams
      Union, intersection and difference
      Partition of a set
 Operations on Sets and Venn Diagrams
There are three basic set operations: union, intersection and difference (relative complement).
The union of two sets A U B is the set that consists of all elements belonging to either set A or set B (or both). We say that an element x is in A U B if either x is in A or x is in B (or x belongs to both).
In formal notation,  A U B = { x x A or x B or both }.
The intersection of two sets A B is the set of all elements that are elements of both A and B
In formal notation,  A B = { x x A and x B }.
                 The union and the intersections of sets A and B represented by Venn Diagrams.
The difference (or the relative complement of B in A), denoted as A B  (or A \ B), is the set of all elements that are elements of A but not of B. In formal notation,  A B = { x x A and x not of B }.
The complement (or the absolute complement) of A in U, denoted as A (or Ac), is the set of all elements of a given universal set that are not elements of A. In formal notation,  A = { x x U and x not of A }.
                      The difference A B and the complement A' represented by Venn Diagrams.
 Example:   Assuming the set of natural numbers N is the universal set given are sets, 
                 A = { x N x < 5 }B = { x N 3 x < 8 } and  C = { x N x = 2n 1, n N }.
   Find:    a)  A B,    b)  A U B,    c)  A B,    d)  B A   and   e)  C'.
Solution:  As  A = { x N x < 5 } = { 1, 2, 3, 4 }, 
                      B = { x N 3 x < 8 } = { 3, 4, 5, 6, 7 }
               and  C = { x N x = 2n 1, n N } = { 1, 3, 5, 7, . . . }
         then  a)  A B = { 3, 4 },    b)  A U B = { 1, 2, 3, 4, 5, 6, 7 },  
                  c)  A B = { 1, 2 },    d)  B A = { 5, 6, 7 } 
         and   e)  C' = { x N x = 2n } = { 2, 4, 6, 8, . . . }. 
Partition of a Set
A partition of a set S is a collection of non-empty disjoint subsets of S, whose union is S. Two sets are said to be disjoint if they have no elements in common.
 Example:  If given set is {1, 2, 3, 4, 5, 6, 7} then one of its possible partitions is { {1, 3, 7}, {2}, {4, 5}, {6} }.
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