0580

Set Notation and Venn Diagrams

Number

How the diagram is built

  • A Venn diagram shows sets as circles inside a rectangle, where the rectangle is the universal set
  • Overlapping circles share the elements that lie in both sets
  • Anything belonging to no named set is written inside the rectangle but outside every circle
  • This specification limits Venn diagrams to two or three sets
Venn diagram with universal set as a rectangle, labelled here with the alternative symbol xi, containing two overlapping circles labelled A and B; 12, 6 and 2 lie in A alone, 14 and 28 in the overlap, 7, 21 and 35 in B alone, and 8, 5 and 1 outside both circles
Source: Set notation by Save My Exams

Regions and operations

  • Each operation corresponds to a region of the diagram, which is what makes the notation easy to apply
  • The union covers both circles entirely; the intersection covers only the overlap
Three Venn diagrams, each a rectangle labelled xi holding two overlapping circles A and B. In the first both circles are shaded, labelled A union B, A or B or both. In the second only the overlap is shaded, labelled A intersection B, A and B. In the third everything inside the rectangle except circle A is shaded, labelled A complement, not A
Source: Set notation by Save My Exams
  • A complement shades everything outside the named set but still inside the rectangle
  • Combined expressions are read from the inside out, so (A ∪ B)′ means shade everything the union does not cover

Counting from a diagram

  • n(A) counts every element inside circle A, including those in the overlap
  • Elements in the overlap belong to both sets and are counted once in a union
Exam tip

Work the expression from the inside outwards: shade the bracket first, then apply a complement last by shading everything left over. The complement of a union is not the union of the complements. Shade firmly and completely — a half-shaded region is ambiguous to a marker.

Worked example

Counting from a Venn diagram

In a group of 30 students, 18 study art and 14 study biology. 7 study both. Work out how many study neither.

Solution:

  • Fill the overlap first, because the two given totals each already include it
  • The intersection holds 7
  • Art only = 18 − 7 = 11
  • Biology only = 14 − 7 = 7
  • Those three regions hold 11 + 7 + 7 = 25 students
  • The universal set holds 30, so the region outside both circles holds 30 − 25 = 5
  • Check by adding all four regions: 11 + 7 + 7 + 5 = 30
  • Writing 18 in the art circle and 14 in the biology circle would count the 7 twice and total 39