Venn diagrams

1 About

This page, at present, is a placeholder. The whole topic of presenting Venn diagrams in a way that is accessible to the blind is potentially hard. I don’t think figure descriptions, however good, will work.

This page sketches, roughly, an alternative presentation.

2 Inclusion-Exclusion (easy)

Use \(\#X\) for the number of elements in a set \(X\). Then the simplest case is \[ \#(A\cup B) = \#A + \#B - \#(A\cap B) \] which I prefer to write as \[ \#(A\cup B) + \#(A\cap B) = \#A + \#B \]

3 Truth table

Suppose we have a finite universe \(U\) with \(n=\#U\). Suppose also that we have subsets \(A_1, A_2, \ldots A_n\) of \(U\). The associated truth table records for each \(u\in U\) and each \(i\) in \(\{1, 2, \ldots, n\}\) whether or not \(u_i \in A_j\). We think of this as a table with \(n\) columns and \(m\) rows.

4 Alternative (easy)

A sighted student, by looking at the Venn diagram, can understand and prove the easy case of inclusion-exclusion. The blind student can perhaps understand and prove this result by looking at the columns of the truth table.

The first key idea is that there only \(4=2^2\) possible values for each column. The second key idea is that for each these possible columns the result holds. The third key idea is that hence the overall result holds.

5 Alternative (hard)

Even for the ordinarily sighted Venn diagrams fail for \(m\) (the number of \(A_j\)) more than about five. In high dimensions we are all blind. In some cases even the sighted student will benefit from the blind approach.

6 Sign errors

I prefer to write the statement of the inclusion-exclusion formula to not have in it any minus signs. I’d like to say more about this, at some time and place.