hmk4sol - COT 3100 Homework #4 Fall 00 Lecturer: Arup Guha...

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COT 3100 Homework #4 Fall ’00 Lecturer: Arup Guha Assigned: 10/17/00 Due: 10/24/00 (No late assignments) 1) Prove or disprove: If a relation R, defined as a subset of A x A, is transitive, then R ° R is transitive as well. We must show that if (a,b) R ° R (b,c) R ° R, that (a,c) R ° R. If we have (a,b) R ° R, by the defn. of relation composition, there exists a d such that (a,d) R (d,b) R. We also know that R is transitive. Hence we can deduce that (a,b) R. If we have (b,c) R ° R, by the defn. of relation composition, there exists a e such that (b,e) R (e,c) R. We also know that R is transitive. Hence we can deduce that (b,c) R. But, if we have that (a,b) R (b,c) R, by defn. of function composition, we know that (a,c) R, as desired. Thus, R ° R is transitive. 2) Define sumdigits(a), where a is a positive integer, to be the sum of the digits in the decimal representation of the positive integer a. Using this definition of the function sumdigits, consider the following relation: R = {(a,b) | a Z + b Z + (a b sumdigits(a) sumdigits(b)} Determine if this relation is (i)reflexive, (ii)irreflexive, (iii)symmetric, (iv)antisymmetric,
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hmk4sol - COT 3100 Homework #4 Fall 00 Lecturer: Arup Guha...

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