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Section 2.4
43
13. (a) [BB] Yes, this is an equivalence relation.
Reflexive: Note that if
a
is any triangle,
a
'"
a
because
a
is congruent to itself.
Symmetric: Assume
a
'"
b.
Then
a
and
b
are congruent. Therefore,
b
and
a
are congruent, so
b", a.
Transitive: If
a
'"
b
and
b
'" c, then
a
and
b
are congruent and
b
and c are congruent, so
a
and c
are congruent. Thus,
a
'" c.
(b) Yes, this is an equivalence relation.
Reflexive:
If
a
is a circle, then
a
'"
a
because
a
has the same center as itself.
Symmetric: Assume
a
'"
b.
Then
a
and
b
have the same center. Thus,
b
and
a
have the same
center, so
b
'"
a.
Transitive: Assume
a
'"
b
and
b
'" c. Then
a
and
b
have the same center and
b
and c have the
same center, so
a
and c have the same center. Thus,
a
'"
c.
(c) Yes, this is an equivalence relation.
Reflexive: If
a
is a line, then
a
is parallel to itself, so
a
'"
a.
Symmetric:
If
a
'"
b,
then
a
is parallel to
b.
Thus,
b
is parallel to
a.
Hence,
b
'"
a.
Transitive:
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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