Nicola Hodges
David Walnut
MATH 290002
November 21, 2010
Writing Assignment 8
1.
Show that the relation R on the natural numbers given by aRb iff b=2
k
a for
some integer k
≥
0 is a partial ordering.
Show that the relation on the natural numbers aRb, where b=2
k
a
for some integer k,j
≥
0 is a partial ordering.
Let a,b,c be natural
numbers.
First show that aRb is reflexive.
So aRa implies, a=2
k
a and
hence 2
k
=1 and k=0. Hence, the relation is reflexive.
Next show that aRb is antisymmetric.
Assume aRb and
bRa and we will show a=b.
So aRb implies b=2
k
a and bRa
implies a=2
j
b.
Rearranging this equation gives b=2
j+k
a which
implies 2
k+j
=1, so k=j=0.
Since k,j
≥
0 then a=b. Therefore the
relation is antisymmetric.
Next show that aRb is transitive.
Assume aRb, and bRc, we
will show aRc.
So aRb implies b=2
k
a and bRc implies c=2
j
b.
Rearranging the equations gives
c=2
k
(2
k
a), c=2
k+j
a, hence aRc.
Therefore, the relation is transitive.
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 Fall '08
 Alligood,K
 Advanced Math, Natural Numbers, Order theory, lower bound, aRb, partial ordering

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