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 Î A or x Î 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 Î A and x Î 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 Î A and x not Î 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 Î U and x not Î 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 Î N x < 5 }B = { x Î N 3 x < 8 } and  C = { x Î N x = 2n 1, n Î N }.
Find:    a)  A B,    b)  A U B,    c)  A B,    d)  B A   and   e)  C'.
Solution:  As  A = { x Î N x < 5 } = { 1, 2, 3, 4 },
B = { x Î N 3 x < 8 } = { 3, 4, 5, 6, 7 }
and  C = { x Î N x = 2n 1, n Î 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 Î 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} }.
Beginning Algebra Contents