**Unit 20 Exercises**

**1. **Let *R* be the parent-child relation on the set of people
that is, *R* = { (*a, b*) | *a* is a parent of *b* }.
Let *S* be the sibling relation on the set of people that is,
*R* = { (*a*, *b*) | *a* and *b* are siblings
(brothers or sisters) }. What are *S ^{o}R* and

**2.** Let *R* be a reflexive relation on a set *A*.
Show that *R ^{n}* is reflexive for all positive intergers

**3. **Let *R* be the relation on the set { 1, 2, 3, 4} containing
the ordered pairs (1, 1), (1, 2), (2, 2), (2, 4), (3, 4), and (4, 1).
Find

a) the reflexive closure of *R*,

b) symmetric closure of *R* and

c) transitive closure of *R*.

**4. **Let *R* be the relation { (*a, b*) | *a*
is a (integer) multiple of *b* } on the set of integers. What is the
symmetric closure of
*R* ?

**5.** Suppose that a binary relation *R* on a set *A* is
reflexive.
Show that *R** is reflexive, where *R** =
*R ^{i}*
.