Venn Diagrams and Set Operations

• Venn diagrams give us a way represent sets.
  • Draw a box representing the universal set (U)
  • Draw each of the two sets.
    • A = B
    • A has nothing in common with b.
    • A is contained in B
    • A and B intersect but A⊄ B and B⊄ A
  • We will use the last because it is most general.
  • Label the regions with roman numerals (I, II, III, IV)
• The complement of set A, written A', is the set of elements in U but not in A.
  • U ={ 1, 2, 3, 4, 5, 6, 7, 8}, A = {1,3,5,7}, A' = {2,4,6,8}
• The intersection of two sets A and B, written A ∩ B, is the set of all elements in both A and B.
  • U ={ 1, 2, 3, 4, 5, 6, 7, 8}, A = {1,3,5,7}, B = {1,2,3,4,5}
  • A ∩ B = { 1, 3, 5}
• The union of two sets A and B, written A ∪ B, is the set of all elements in either set A or set B (or both)
  • U ={ 1, 2, 3, 4, 5, 6, 7, 8}, A = {1,3,5,7}, B = {1,2,3,4,5}
  • A ∪ B = { 1, 2, 3, 4, 5, 6, 8}
• The difference of two sets A and B, written A-B is the set of all elements in set A which are not in set B.
  • U ={ 1, 2, 3, 4, 5, 6, 7, 8}, A = {1,3,5,7}, B = {1,2,3,4,5}
  • A - B = { 7 }
• The number of elements in A ∪ B
  • This is not a "natural" instinct for some.
  • n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
  • Reason this out with a Venn diagram.