The concept of a function is one of the most important in mathematics. Most participants were familiar with the idea of a function as something that has inputs and outputs. We took some time to restate this idea, describe some terminology surrounding functions, and extend the idea to multivariate functions. One of the key uses of functions in mathematics is in defining the cardinality of sets: Two sets are said to be of the same size if there is a function that determines a one-to-one correspondence between their elements.

We then investigated a single, very special function called Cantor’s pairing function. It is a function of two variables and has the formula f(x,y)=(1/2)(x+y)(x+y+1)+y. While it is hard to visualize what the function does just from the formula, we explored what the function does using test values. We eventually discovered the unique role this function plays in the theory of set cardinality.