COT3100Hmk06 - productdigits consider the following...

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COT 3100 Fall 2002 Homework #6 Assigned: 11/13/02 (Wednesday) Due: 11/26/02 (Tuesday) in lecture 1) Let f : A B and g: B C denote two functions. If both f and g are surjective, prove that the composition g ° f: A C is a surjection as well. 2) Let g: A A be a bijection. For n 2, define g n = g ° g ° ... ° g, where g is composed with itself n times. Prove that for n 2, that (g n ) -1 = (g -1 ) n . 3) Prove or disprove: Let R be a relation over A x A. If R ° R is transitive, then R is transitive as well. 4) Prove that the following relation is an equivalence relation, and determine how many equivalence classes R partitions the set Z + into. R = {(a,b) | a Z + b Z + 10 | (a 2 - b 2 )} 5) Define productdigits(a), where a is a positive integer, to be the product of the digits in the decimal representation of the positive integer a. Using this definition of the function
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Unformatted text preview: productdigits, consider the following relation: {(a,b) | a ∈ Z + ∧ b ∈ Z + ∧ (a ≥ b ∨ productdigits(a) ≥ productdigits(b)} Determine if this relation is (i)reflexive, (ii)irreflexive, (iii)symmetric, (iv)antisymmetric, and (v)transitive? 6) Prove that following function is a bijection from the open interval (0,3) to the positive real numbers: x x x f 3 3 ) (-= For the last few questions assume that R and T are arbitrary binary relations over ZxZ. Disprove each of the following assertions: (Note: r, s, and t represent the reflexive, symmetric and transitive closures of relations.) 7) If s( R) is transitive, then R is transitive. 8) If R is transitive and R ⊆ T , then T is transitive. 9) For all relations R , t(s( R )) is reflexive. 10) For all relations R, t(s( R )) = s(t( R ))....
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