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Unformatted text preview: a,b,c,d,n âˆˆ Z , k âˆˆ N with a â‰¡ b (mod m ) and c â‰¡ d (mod m ). Then (a) a + c â‰¡ b + d (mod m ) (b) ac â‰¡ bd (mod m ) 1 (c) ac â‰¡ bd (mod m ) (d) a k â‰¡ b k (mod m ) (e) Find the remainder when 6 2009 is divided by 37. 7. Let R be a relation on a set A . (a) Prove that R is transitive if and only if R 2 âŠ† R . (b) Let 1 A = { ( x,x ) : x âˆˆ A } ( 1 A is called the identity relation on A ). Prove that R is reï¬‚exive if and only if 1 A âŠ† R . 8. Suppose that R is a relation on a set A . Is RR1 the same as 1 A ? Justify your answer. 9. Let R be a relation from A to B and let 1 B be the identity relation on B . Prove that R 1 B = R . 2...
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 Spring '10
 DrChanSongHeng
 Equivalence relation, Binary relation, Transitive relation, relation, Transitive closure, FOUNDATION OF MATHEMATICS

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