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3. Relations |
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1. Define two points and of the plane to be
equivalent if . Check that this is an equivalence
relation and describe the equivalence classes.
Proof. $\quad$ It is easily seen that the relation is
reflexive, symmetric, and transitive. Each equivalence class is a parabola
given by .$\quad\square$
2. Let be a relation on a set . If , define the
restriction of to to be the relation .
Show that the restriction of an equivalence relation is an equivalence
relation.
Proof. $\quad$It is clear that:
$$
\forall x\in A_0\forall y\in A_0((x,y)\in C
\Leftrightarrow (x,y)\in C\cap (A_0\times A_0)).
$$
Thus all the properties for an equivalence relation hold in
.$\quad\square$
3. Here is a “proof” that every relation that is both symmetric and
transitive is also reflexive: “Since is symmetric, implies .
Since is transitive, and together imply , as desired.”
Find the flaw in this argument.
Proof. $\quad$ Let be a relation.
If is
symmetric and transitive, then:
$$
\forall a\forall b(aCb\Rightarrow aCa).
$$
If is reflexive, then:
$$
\forall a\in A(aCa).
$$
4. Let be a surjective function. Let us define a relation on
by setting if .
$\quad$(a) Show that this is an equivalence relation.
$\quad$(b) Let be the set of equivalence classes.
Show there is a bijective correspondence of with .
Proof. $\quad$(a) . and
. .
$\quad$(b) Let be a function given by
where , and let
and .
Since , , thus is well-defined.
If and and , then , thus
, so is injective. Since is surjective, for every
there is such that , and since is
an equivalence relation, there is such that ; , so is surjective. Thefefore, is
bijective.$\quad$
5. Let and be the following subsets of the plane:
$$
\begin{array}{rl}
S &={(x, y)\mid y = x + 1\text{ and }0 < x < 2},\
S' &={(x, y)\mid y - x\text{ is an integer}}.
\end{array}
$$
$\quad$(a) Show that is an equivalence relation on the real
line and . Describe the equivalence classes of .
$\quad$(b) Show that given any collection of equivalence relations
on a set , their intersection is an equivalence relation on .
$\quad$(c) Describe the equivalence relation on the real line
that is the intersection of all equivalence relations on the real line that
contain . Describe the equivalence classes of .
Proof. $\quad$(a) . .
for .
$\quad$(b) Let be the collection of
equivalence relations on indexed by a nonempty set .
For all , since for each , . If , then for each , thus
for each , so
. Similarly, if , then .
$\quad$(c) A equivalence relation on the real line that contain
need more equations. for the reflexivity, for the symmetry.
Thus and . for the
transitivity, thus in general, is an integer, and .
is the restriction of to . This definition is minimal
with respect to the previous equations. can be seen as the intersection
of two equivalence relations,
Extra open brace or missing close brace
$S'\cap
{(x,y)\mid\text{either }0<x<3$
and , or or
and or
Extra close brace or missing open brace
$y\ge 3)}$
.
6. Define a relation on the plane by setting
$$
(x_0, y_0) < (x_1, y_1)
$$
if either , or
and . Show that this is an order relation on the plane, and
describe it geometrically.
Proof. $\quad$It is easily seen that comparability,
nonreflexivity and transitivity hold for the given relation.
Geometrically, if , then
lies in for some
and lies in for some and
either , or and .$\quad\square$
7. Show that the restriction of an order relation is an order relation.
Proof. $\quad$Let and be sets such that
and let be an order relation on . It is clear that:
$$
\forall x\in A_0\forall y\in A_0((x,y)\in C
\Leftrightarrow (x,y)\in C\cap (A_0\times A_0)).
$$
Thus all the properties for an order relation hold in
.$\quad\square$
8. Check that the relation defined in Example 7 is an order relation.
Proof. $\quad$From Example 7, "Define if ,
or if and ."
$\quad$Clear.$\quad\square$
9. Check that the dictionary order is an order relation.
Proof. $\quad$Clear.$\quad\square$
10. $\quad$(a) Show that the map of
Example 9 is order preserving.
$\quad$(b) Show that the equation
defines a function that is both a left and a right
inverse for .
Proof. $\quad$(a) From Example 9, is given by . Suppose that .
. Since and ,
. Thus is order preserving; thus injective, and also neither
upper-bounded nor lower-bounded; thus surjective. Therefore,
and have the same order type.
$\quad$(b) Brute-force is enough. ;-)$\quad\square$
11. Show that an element in an ordered set has at most one immediate
successor and at most one immediate predecessor. Show that a subset of an
ordered set has at most one smallest element and at most one largest element.
Proof. $\quad$Let be an ordered set, and
let .
If has immediate successors, and , then by comparability,
; otherwise or , a contradiction. Similarly to
immediate predecessor, smallest element, and largest element.$\quad\square$
12. Let $\mathbb{Z}+$ denote the set of positive integers. Consider
the following order relations on $\mathbb{Z}+\times\mathbb{Z}_+$:
$\quad$(i) The dictionary order.
$\quad$(ii) if either , or and .
$\quad$(iii) if either ,
or and .
$\quad$In these order relations, which elements have
immediate predecessors? Does the set have a smallest element?
Show that all three order types are different.
Proof. $\quad$(i) for every $x\in\mathbb{Z}+$
has no immediate predecessor. $(1,1)$ is the smallest.
$\quad$(ii) $(x,1)$ and $(1,y)$ for every
$x,y\in\mathbb{Z}+$ have no immediate predecessor. No smallest element.
$\quad$(iii) has no immediate predecessor.
is the smallest.
$\quad$ Suppose , where
is bijective order preserving function, and has
immediate predecessor
, and has no immediate predecessor, then there is no .
They all have different order types.$\quad\square$
13. Prove the following:
Theorem. If an ordered set A has the least upper bound property,
then it has the greatest lower bound property.
Proof. $\quad$Let be bounded below, and
let
Extra open brace or missing close brace
$T={x\in A\mid x$
is a lower bound
of
Extra close brace or missing open brace
$S}$
be nonempty. has a least upper bound , and clearly
is a greatest lower bound of .$\quad\square$
14. If is a relation on a set , define a new relation on
by letting if .
$\quad$(a) Show that is symmetric if and only if .
$\quad$(b) Show that if is an order relation, is also an
order relation.
$\quad$(c) Prove the converse of the theorem in Exercise 13.
Proof. $\quad$(a) Clear.
$\quad$(b) "If and , then ." implies
"If , then ". is transitive; the other properties
are obvious.
$\quad$(c) Let be bounded above and let
Extra open brace or missing close brace
$T={x\in A\mid x$
is a upper bound of
Extra close brace or missing open brace
$S}$
be nonempty.
has a greatest lower bound
, and clearly is a least upper bound of .$\quad\square$
15. Assume that the real line has the least upper bound property.
$\quad$(a) Show that the sets
$$
\begin{array}{ll}
\left[0,1\right] &={x\mid 0\le x\le 1},\
\left[0,1\right) &={x\mid 0\le x <1}
\end{array}
$$
have the least upper bound property.
$\quad$(b) Does in the dictionary order have the
least upper bound property? What about ? What about
?
Proof. $\quad$(a) Let . Since of
is bound above, has a least upper bound .
Clearly , thus also has the least upper bound property.
Let . Since of is
bounded above, has a least upper bound such that .
If , then of is not bounded above. Thus
if of is bounded, then . Therefore, has the least
upper bound property.
$\quad$(b) and have
the least upper bound property.
has no least upper
bound.$\quad\square$