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Unformatted text preview: a,b,c,d,n ∈ Z , k ∈ N with a ≡ b (mod m ) and c ≡ d (mod m ). Then (a) a + c ≡ b + d (mod m ) (b) ac ≡ bd (mod m ) 1 (c) ac ≡ bd (mod m ) (d) a k ≡ 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 ⊆ R . (b) Let 1 A = { ( x,x ) : x ∈ A } ( 1 A is called the identity relation on A ). Prove that R is reﬂexive if and only if 1 A ⊆ 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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This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.
 Spring '10
 DrChanSongHeng

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