(a) Let V = {1,2,3,4,5,6,7} and let E = {(1,7), (2.1). (4.1), (6,5), (6,6)} be a...

  

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(a) Let V = {1,2,3,4,5,6,7} and let E = {(1,7), (2.1). (4.1), (6,5), (6,6)} be a binary relation on V. Find the equivalence closure of this relation and state the equivalence classes. (It may help to draw a diagram.) [4 marks] (b) The coefficients of the following system are taken from GF (2). Solve it using Gaus- sian elimination. X₁ + x₂ + x4 = 0 X₁ + x3 + x4 =1 X2 + X5 = 0 X1 + x2 + x3 + X5 = 0 X₁ + X3 = 0 [5 marks] (c) Let S be the smallest subset of {a,b}* such that all of the following conditions are satisfied: • EES. • If w€ S, then aabw € S. • If w€ S, then any anagram of w is also in S. (An anagram of w is a string that arises from w by a permutation of the letters.) (i) Show that aabababaa is in S. [2 marks] (ii) Argue that ababab is not in S by giving a property and proving that all elements of 5 satisfy this property. [5 marks] (d) Consider the following Java methods. Do they represent functions? If yes, are they injective, surjective, or bijective? Justify your answers. int doubleInt(int number) { return number * 2; } float addOneToFloat (float number) { return number + 1.0; } [4 marks] (a) Let V = {1,2,3,4,5,6,7} and let E = {(1,7), (2.1). (4.1), (6,5), (6,6)} be a binary relation on V. Find the equivalence closure of this relation and state the equivalence classes. (It may help to draw a diagram.) [4 marks] (b) The coefficients of the following system are taken from GF (2). Solve it using Gaus- sian elimination. X₁ + x₂ + x4 = 0 X₁ + x3 + x4 =1 X2 + X5 = 0 X1 + x2 + x3 + X5 = 0 X₁ + X3 = 0 [5 marks] (c) Let S be the smallest subset of {a,b}* such that all of the following conditions are satisfied: • EES. • If w€ S, then aabw € S. • If w€ S, then any anagram of w is also in S. (An anagram of w is a string that arises from w by a permutation of the letters.) (i) Show that aabababaa is in S. [2 marks] (ii) Argue that ababab is not in S by giving a property and proving that all elements of 5 satisfy this property. [5 marks] (d) Consider the following Java methods. Do they represent functions? If yes, are they injective, surjective, or bijective? Justify your answers. int doubleInt(int number) { return number * 2; } float addOneToFloat (float number) { return number + 1.0; } [4 marks]

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a Equivalence Closure and Equivalence Classes Final Answer Equivalence Closure 17 2 1 4 1 6 5 6 6 1 ... View the full answer