The relation **nhbr**, which we're defining as a set (of pairs), could also be thought of being a function. We say that the **indicat or**
**function** of a set is a Boolean function indicating whether its input is in the set or not. So instead of being given the set **nhbr**,
you would have been equally happy with its indicator function **f**_{nhbr}, where (for example) **f**_{nhbr} (*B*, *C*) = **true** and **f**_{nhbr} (*B*, *Q*) = **false**. Similarly, if you know that **f**_{hasPirate} (*K*) = **true** and that **f**_{hasPirate} (*L*) =
**false**, then this is enough information to conclude that and .
**The set and the function are equivalent ways of modeling the same underlying relation.**

The next two exercises aren't meant to be difficult, but rather to illustrate that, while we've sketched these two approaches and suggested they are equivalent, we still need an exact definition.

**Exercise 3.2.1**

For the indicator function on the domain of (pairs of) natural numbers, write down the set-of-pairs representation for the corresponding binary relation. It's insightful to give the answer both by listing the elements, possibly with ellipses, and also by using set-builder notation. In general, for a binary indicator function f, what, exactly, is the corresponding set?

**Exercise 3.2.2**

For the relation **hasPirate = { K, T, R, U, E}** on the set of (individual) WaterWorld locations, write down the
indicator-function representation for the corresponding unary relation.

**In general**, how would you write down this translation?

Since these two formulations of a relation, sets and indicator functions, are so close, we'll often switch between them (a very slight abuse of terminology).

Think about when you write a program that uses the abstract data type Set. Its main operation is `elementOf`. When might you use an explicit enumeration to encode a
set, and when an indicator function? Which would you use for the set of WaterWorld locations? Which for the set of prime numbers?

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