**Unit 19 Exercises**

**1.** For each of the following relations on the set {1, 2, 3, 4},
decide whether it is reflexive, symmetric, antisymmetric and/or transitive.

a) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

b) {(1, 3), (1, 4), (2, 3), (3, 4)}

c) {(1, 1), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 3), (3, 4)}

**2.** Determine whether the relation *R* on the set of all
integers is reflexive, symmetric, antisymmetric, and/or transitive,

where (*x*,
*y*) *R* if and only if

a) *x* is divisible by *y*.

b) *x *
*y.*

c) *y = x + 2 * or
*y = x - 2.*

d) *x* = *y*² + *1*.

**3. **Let *A* be the set of people in your town.
Let *R*1 be the unary relation representing the people in your town
who were registered in the last election and *R*2
be the unary relation representing the people in your town
who voted in the last election.
Describe the 1-tuples in each of
the following relations.

a) *R*1
*R*2.

b) *R*1
*R*2.

**4. **Draw the directed graph
that represents the relation {(*a*,*b*), (*a*, *c*),
(*b*, *c*), (*c*, *b*), (*c*, *c*),
(*c*, *d*), (*d*, *a*), (*d*, *b*)}.