birthday problem

The birthday problem is an example of how we could apply the pigeonhole principle.

Initially we have:

Annie, Batul, Charlie, Deja, Evelyn, Fawwaz, Gregoire, and Hoon talk to each other and discover that Deja and Gregoire were both born on Tuesdays.

After some simplification:

In any group of eight people, some two of them were born on the same day of the week.

It turns out that we can convert it further, into the language of set theory:

P = {Annie, Batul, Charlie, Deja, Evelyn, Fawwaz, Gregoire, Hoon} D = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}.

Now we can see that $|P|>|D|$, and there are at least two elements in $P$ mapped to the same element in $D$ by the pigeonhole principle.