Lesson 7 of 11 · 8 min
Universal set and Venn diagrams
NCERT §1.7, §1.8
Every question about the sign-up sheet is about roll numbers 1 to 20. Nobody asks whether roll 35 plays cricket. That background set has a name.
The lesson in notes
In short
In any discussion there is a basic set that contains all the sets being talked about; it is the universal set U. Its choice depends on the context: for work with integers, U may be Q or R.
A Venn diagram, named after the English logician John Venn (1834-1923), draws U as a rectangle and its subsets as circles (or other closed curves) inside it, with elements written in the regions they belong to.
B ⊂ A is drawn with B's circle entirely inside A's circle; sets with no common element are drawn as circles that do not overlap.
A proposed universal set must contain every element of every set in the problem: for {1, 3}, {2, 4} and {0, 4, 8}, the set {0, 1, 2, …, 8} can serve as U, but {1, 2, 3, 4} cannot and φ never can.
Two overlapping circles split U into four regions: in A only, in both, in B only, and in neither. Every union, intersection, difference and complement of two sets is a choice of some of these four regions.