More complicated question on Venn diagrams involve the use of simultaneous equations, or the introduction of a dummy variable x.

ExampleSketch the Venn diagram.

(1)

(2)

Divide (1) by (2) to get

Substitute these into the equation

Thenand from (1)

We can draw the Venn diagram.

Example:andare independent.andFind the possible values of and draw a possible Venn diagram.

Label the intersectionthenandSinceandare independent

or

A possible diagram is shown below.