Math 210 Additional Problems Due 9/21/15 For the next two problems, if u and v are integers we say u divides v, and write u|v, if there exists and integer k such that v = ku. 1) Let a and b be integers and let A = { x Z | a| x } and B = { x Z | b| x }. Find and prove a necessary and sufficient condition for A B. 2) Let c and d be integers and let C = { x Z | x |c} and D = { x Z | x |d}. Find and prove a necessary and sufficient condition for C D. 3) State and prove necessary and sufficient conditions for A \\ B = B \\ A 4) Prove the following about set complements. Here the letters A, B, and C denote subsets of the universe U. a) A = B if and only if Ac = Bc b) ( Ac )c = A c) ( A B C )c = Ac Bc C c 5) Give an example of relation on a set that is both transitive and symmetric, but not reflexive. Then explain what is wrong with this proof: Statement: If R is symmetric and transitive, then it is reflexive. p f : Suppose R is symmetric and transitive. Symmetric means that xRy yRx. We apply transitivity to give xRx. Therefore R is reflexive. 1