assn4dfall2009 - a 4 b 4 (mod 30) on the set { 1 , 2 , 3 ,...

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CONCORDIA UNIVERSITY COMP 232/2 Mathematics for Computer Science FALL 2009 Assignment 4 1. Give the directed graph of the relation ± ( a, b ) ² ² a 2 - b 2 8 (mod 32) ³ on the set { 3 , 5 , 7 , 11 , 13 , 17 , 19 } . 2. For a function f from the set A to the set B we define the graph of f to be the set ± ( a, b ) ² ² a A , b B , b = f ( a ) ³ . Let R be a relation from A to B and let Δ A and Δ B denote the diagonal relations on A and B respectively. Show the following. i) R is the graph of a function from A to B if and only if Δ A R - 1 R and R R - 1 Δ B . ii) R is the graph of an invertible function from A to B if and only if Δ A = R - 1 R and R R - 1 = Δ B . 3. Find all relations on the set { 1 , 2 , 3 } with the following combinations of properties. (Give your answers in matrix form.) i) reflexive and symmetric but not transitive ii) reflexive and symmetric and transitive but not antisymmetric iii) symmetric and transitive but not reflexive and not antisymmetric 4. Give the equivalence classes of the equivalence relation ± ( a, b ) ²
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Unformatted text preview: a 4 b 4 (mod 30) on the set { 1 , 2 , 3 , . . . , 15 } . 5. Let A 3 = { 1 , 2 , 3 } and let A 4 = { 1 , 2 , 3 , 4 } . Determine whether the following are partial orders. i) A 3 , { (1 , 1) , (1 , 3) , (2 , 1) , (2 , 2) , (3 , 3) } ii) A 3 , { (1 , 1) , (2 , 2) , (3 , 1) , (3 , 3) } iii) A 4 , { (1 , 1) , (1 , 3) , (2 , 2) , (2 , 3) , (3 , 3) , (3 , 4) , (4 , 1) , (4 , 2) , (4 , 4) } 6. Let N be the set of nonnegative integers and let T 1 , T 2 , T 3 and T be the relations on N N given by T 1 = ( ( a, b ) , ( c, d ) ) a + b < c + d , T 2 = ( ( a, b ) , ( c, d ) ) a + b = c + d , T 3 = ( ( a, b ) , ( c, d ) ) a c , T = T 1 ( T 2 T 3 ) . i) Show that T is a partial order on N N . ii) Is T a total order on N N ? Explain your answer. 1-Dec-2009, 1:50 am...
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