44
Solutions to Exercises
16. (a) [BB] The given statement is an implication which concludes
"x

y
=
x

y,"
whereas what is
required is a logical argument which concludes "so
rv
is reflexive."
A correct argument is this: For any
(x,y)
E R2,
x

Y
=
x

y;
thus,
(x, y)
rv
(x, y).
Therefore,
rv
is reflexive.
(b) There is confusion between the elements of a binary relation on a set
A
(which are ordered pairs)
and the elements of
A
which are themselves ordered pairs in this situation. The given statement
is correct
provided
each of
x
and
y
is understood to be an ordered pair of real numbers, and we
understand
n
=
{(x,
y)
I
x
rv
y}
but this is very misleading. Much better is to state symmetry
like this:
if
(x, y)
rv
(u, v),
then
(u, v)
rv
(x, y).
(c) The first statement asserts the implication
"x

y
=
u

v
~
(x,y)
rv
(u,v)"
which is the
converse of what should have been said. Here is the correct argument:
If
(x, y)
rv
(u, v),
then
x

y
=
u

v, sou

v
=
x

y
and, hence,
(u, v)
rv
(x, y).
(d) This suggested answer is utterly confusing. Logical arguments consist of a sequence of implica
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 Summer '10
 any
 Logic, Graph Theory, Binary relation, Cartesian product, Transitive relation, equivalence class, iy

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