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# WA 8 - Nicola Hodges David Walnut MATH 290-002 Writing...

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Nicola Hodges David Walnut MATH 290-002 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 anti-symmetric. 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 anti-symmetric. 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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