\(\newcommand{\set}[1]{\{\, #1 \,\}}\)

Problem 1: Relatable

Consider the following artificial set:

\[ S = \{\, a, b, c, d, e, f \,\} \]

As well as the following relation \(R\) over \(S\):

\[ R = \{\, (a, b), (a, c), (b, d), (c, c), (d, b), (e, b), (e, f) \,\} \]

1. Compute the following operations over \(R\):

   • \(\mathrm{dom}(R)\).
   • \(\mathrm{range}(R)\).
   • \(R^{-1}\).
   • \(R(a)\).

2. Instantiation this artificial example to a real-life example. That is, give real-life meaning to the set \(S\) and the relation captured by \(R\). Describe this real-life example in a sentence or two.

3. For each of the operations from part (a), interpret what the operation is “computing” in terms of your real-life example.