1
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(i)
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Impossible.
If we were allowed to have a empty domain, the combination would
be possible. But in a non-empty domain no reflexive relation can
be asymmetric. For, suppose that the relation is expressed by Rxy,
and that that a is one of the things is the domain. Then, if it
is reflexive, Raa is true. But in that case it is not true that
"x"y[Rxy®¬Ryx], because it is not true that [Raa®¬Raa].
To put it another way: if a relation is reflexive, and the domain
is not empty, there are loops in its graph; but loops are double
arrows; so it is not asymmetric.
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(ii)
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This
is possible as long as the relation is empty. If the relation is
not empty it is not possible. For, suppose such a relation is expressed
by Rxy. If it is not empty it must be that (say) Rab (where a and
b may or may not be the same thing). Since the relation is symmetric
it must also be that Rba. But then, since the relation is transitive,
it follows that Raa. ([[RabÙRba]®Raa]). So it is not irreflexive. (If it is
not empty there is an arrow from a to b. Since it is symmetric the
arrow is double. So there is an arrow from a to b, and an arrow
from b to a. So there is a short cut from a to a. So the relation
is not irreflexive.)
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(iii)
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Impossible.
For, suppose such a relation is expressed by Rxy. Since it is non-symmetric,
it must be that (say) Rab and Rba (where a and b may or may not
be the same thing). Since the relation is transitive, it follows
that Raa must be true. (So must Rbb.) So the relation is not irreflexive.
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(iv)
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This
is possible. For instance the relation expressed by x>y fits
the bill.
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(v)
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This
is impossible. For, suppose such a relation is expressed by Rxy.
Since it is non-reflexive, it must be that Raa (say) and ¬Rbb (say),
where a and b must, to course, be different. Since it is connected,
therefore, either Rab or Rba must be true. But since it is symmetric,
if either is true, so must the other be; so Rba and Rab. But then,
if it is transitive, Rbb must be true. But it isn't.
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(vi) |
This is
possible, as long as the relation is empty and the domain has just
one member - otherwise, because it is connected, the relation can't
be empty.
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(vii) |
This
is possible. An example would be a relation whose graph contained
two dots with a loop on one and a single arrow from one to the other.
(For an actual example one could always draw such a graph, and let
the domain consist of the dots in it, and let the relation hold
between x and y just if there is an arrow from x to y.)
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2
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(i) |
Irreflexive;
symmetric; non-transitive; not connected. (It is not transitive,
because there are some broken journeys without short cuts: suppose
that John is Mary's brother; then Mary is John's sister; so there
is a broken journey from John to John, but John is not his own brother
or sister.)
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(ii) |
Irreflexive; symmetric; intransitive; not connected.
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(iii) |
Irreflexive;
asymmetric; both transitive and intransitive; not connected. (It
is both transitive and intransitive because there are no broken
journeys, since if a is b's wife, b must be a man, and therefore
no one's wife.)
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(iv) |
Irreflexive; asymmetric; transitive; not connected.
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(v) |
Reflexive;
non-symmetric; non-transitive; connected. (It is not transitive:
suppose that John is four inches taller than Peter, and that Peter
is four inches taller than James, then John is eight inches taller
than James; so he is at least six inches taller than him.
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(vi) |
This relation
is empty in the given domain (no one being more than ten feet tall).
So it is: irreflexive; symmetric and asymmetric; transitive and
intransitive; not connected.
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(vii) |
This
holds between x and y whatever x and y are. So it is: reflexive;
symmetric; transitive; connected.
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(viii) |
Let
us suppose that it is true that John plays in the same team as Mary
as long as there is a team they both play in, even if there is a team
that Mary plays in but John does not play in. Then the relation is:
non-reflexive (not reflexive, because some people do not play in any
teams at all); symmetric; non-transitive (not transitive, because,
given what we are supposing, John may play in the same team as Mary,
and Mary in the same team as Peter, without its being true that John
plays in the same team as Peter, if Mary plays in different teams);
not connected.
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3
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(i) |
Irreflexive;
symmetric; non-transitive; connected.
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(ii) |
The
relation is empty in this domain; so it is: irreflexive; symmetric
and asymmetric; transitive and intransitive; connected.
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(iii) |
Irreflexive;
symmetric; intransitive; connected. (It is intransitive because,
given that there are only two things in the domain, if x is not
the same person as y and y is not the same person as z, x must be
z.)
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