Set Theory: Drawing Venn Diagrams

Sometimes you may be given the description of some sets and you are asked to draw a Venn diagram to illustrate the sets.

First, we need to determine the relationships between the sets such as subsets and intersections. There could be several ways to describe the relationships.

We would draw A within B if we know that:

All members of A belongs to B or A ⊂ B or A ∪ B = B or A ∩ B = A

or n(A ∩ B) = n(A)

We would draw A overlap B if we know that:

Some members of A belongs to B or A ∩ B ≠ Ø or n(A ∩ B ) ≠ 0

We would draw disjoint sets A and B if we know that

No members of A belongs to B or A ∩ B = Ø or n(A ∩ B ) = 0

Example:
U = the set of triangles, I = the set of isosceles triangles,

Q = the set of equilateral triangles and R = the set of right-angled triangles.

Draw a Venn diagram to illustrate these sets.

Solution:
First, we determine the relationships between the sets.

All equilateral triangles are isosceles, so Q ⊂ I. (within)

Some right-angled triangles may be isosceles. R ∩ I ≠ Ø (overlap)

Right-angled triangles can never be equilateral. R ∩ Q = Ø (disjoint)

Then we draw the Venn diagram:

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