7.1 RELATIONS 233numbers x, y, and z. The relation is antisymmetric, although only by
means of a logical technicality. The question is whether the statement
"for all x, y E R, if x < y and y c x, then x = y" is true. The answer is
"yes" since the premise "x < y and y < x" is false for any x and y. 0EXAMPLE 8 Let A = {1,2,3,4). Let R be a relation on A defined by R =
((1, I), (2,2), (3,319 (4,4), (1,2), (1,3), (1,4), (2,4)). You should check, by
dealing with all possible cases, that R reflexive, antisymmetric, and
transitive, and is not symmetric.The four properties in Definition 3 are dealt with further in the exercises
that follow, as well as throughout the remainder of this chapter.
DEFINITION 4
Let R be a relation from a set A to a set 6. We define the domain of R to be
the set dorn R = (x E A~X R y for some y E 6) and the range of R by the rule
rng R= {YE BlxRyfor some XEA}.Clearly whenever R is a relation from A to B, dorn R is a subset of A and
rng R is a subset of B. In fact, dorn R consists of those elements of A that
are first elements of ordered pairs in R, whereas rng R comprises the ele-
ments of B that are second elements of ordered pairs in R.
EXAMPLE 9 Using the sets and relations defined in Example 1, we note
that dorn R, = (1,2,3) = A, whereas rng R, = (x, y, z) G B. Also,
dorn R, = (21, whereas rng R, = B. Borrowing from Example 3, we
observe that dorn R, = H (since everyone has biological parents), but
rng R, c H (since not everyone & a parent).
DEFINITION 5
Let R be a relation from a set A to a set B. We define the relation R inverse,
denoted R-', by the rule R-' = {(y, x) I(x, y) E R}.Clearly R- is a relation from B to A; it is gotten from R by flipping
over all the ordered pairs in R. We gather together other properties of
R-I in the next theorem.
THEOREM 3
Let R be a nonempty relation from A to B. Then:
(a) dorn (R- ') = rng R
(b) rng (R-') = dorn R
(c) R = (R-')-'
(d) R = R-' if and only if A = B and R is symmetric
(e) R n R-' = I, if and only if A = B and R is antisymmetric