Recently at the Girls’ Angle Club, we mentioned that from a contradiction, one can prove anything. One of our members, Barry Allen (pseudonym!), challenged us to prove that potatoes are made of ducks given that potatoes are made of wood.

So we did.

Before giving our “proof”, please note that for the statement “potatoes are made of wood” to be a contradiction, it must also be true that “potatoes are not made of wood”.  That potatoes are not made of wood, was the standing assumption that allowed for the statement “potatoes are made of wood” to be regarded as a contradiction.

Now to our proof.

Let W be the set of food items that are made of wood. Let N be the complement of W, that is, the set of all food items that are not in W.

Let s be the size of the intersection of W and N.

By definition of W and N, no food item is in both W and N.  Therefore, s = 0.

Potatoes are not made of wood, and therefore, potatoes are not in W, and, hence, are in N.  On the other hand, we are assuming that potatoes are in W.  Therefore, potatoes are in both W and N and therefore the intersection of W and N contains at least one food item (namely, the potato).  Therefore, s > 0.

However, we also know that s = 0.

Since s > 0, we can divide the equation s = 0 by s on both sides to arrive at the equation

1 = 0.

It is well-known that potatoes are not made of ducks.  In particular, potatoes contain 0 ducks.  However, 1 = 0, so if potatoes contain 0 ducks, then potatoes contain 1 duck.

Furthermore, if potatoes contain 1 of anything else, then, since 1 = 0, potatoes contain nothing of anything else.

We conclude that potatoes contain 1 duck and nothing else.  Therefore, potatoes are made of ducks.