Partitions, Surjections, and Equivalence Relations – Benjamin Friedman (February 5, 2019)

One idea being communicable in multiple different ways is a recurring
phenomenon in mathematics. This talk will explore an accessible example.
Namely, we will discuss three ideas (see below), which on the surface, seem
somewhat different. However, we will reveal that all three communicate
the same basic concept, and justify this with accessible examples.

  1. A partition of a set X is a collection of subsets \{A_i\}_{i\in I} that is pairwise disjoint:

        \[A_i\cap A_j=\emptyset, \text{~when~}i\neq j, \]

    and whose union is X:

        \[\bigcup_{i\in I} A_i=X. \]

  2. An equivalence relation on a set X is a relation \sim that is reflexive,
    symmetric, and transitive.
  3. A surjection f:X\to Y is a function whose range is equal to its
    codomain. That is \forall y\in Y, \exists x\in X such that f(x)=y.