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Tutorial 1 : Relations

Tutorial 1 : Relations. Exercise 3Let X = {a, b, c, d, e}. Consider a partition of X consisting of the subsets A = {a, b, e} and B= {c, d}.Write down the equivalence relation ~ induced by this partition (x ~y iff x and y belong to the same subset), using infix notation to give the orde

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Tutorial 1 : Relations

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    1. Tutorial 1 : Relations Exercise 1 Let A= {a, b, c} and B = {1,2}. Give an example of a relation from A to B, give the domain and the range of this relation, and draw an arrow diagram representing it. Exercise 2 Let A = {a, b, c, d}. Consider the relation R = {(a, a), (a, b), (a, c), (b, b), (b, a), (c, a), (d, d)} 1- Draw a digraph representing the relation. 2- Is R reflexive? Is it symmetric? Is it transitive? (Give reasons to your answer)

    2. Tutorial 1 : Relations Exercise 3 Let X = {a, b, c, d, e}. Consider a partition of X consisting of the subsets A = {a, b, e} and B= {c, d}. Write down the equivalence relation ~ induced by this partition (x ~y iff x and y belong to the same subset), using infix notation to give the ordered pairs.

    3. Tutorial 1 : Relations Exercise 4 Let Y = {-2, 1, 2, 3, 6}. Let S ? Y × Y be given by (x, y) ? S if x | y. Show that S is not antisymmetric, and therefore is not a partial order. Exercise 5 Let X = {2, 4, 5, 10, 15, 20} and let R ? X × X be given by (x, y) ? R if x | y. 1. Show that R is a partial order. 2. Draw the digraph and the Hasse diagram of this relation.

    4. Tutorial 1 : Relations Exercise 6 Let X be the powerset of {1, 2}, so that X = {?, {1}, {2}, {1, 2}}. Consider the relation ? on X, where A ? B iff every member of A also belongs to B.  1. Is ? a partial order relation? 2. Is R linear? (explain your answer) 3. Build a linear order ?1 that is a topological sorting of ?.

    5. Tutorial 1 : Relations Exercise 8 Let X = {a, b, c} and let = be the usual alphabetic order relation on X, so that a = b and b = c and of course reflexivity and transitivity hold, giving the pairs a = a, b = b, c = c, a = c as well. List the pairs in the lexicographic order relation ? on X × X.

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