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MAS111_09_assignment_10_hints

# MAS111_09_assignment_10_hints - MAS 111 FOUNDATION OF...

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MAS 111 FOUNDATION OF MATHEMATICS ASSIGNMENT 10 HINTS 1. Here are some relations on a set A = { 1 , 2 , 3 , 4 , 5 } . Determine for each relation whether it is reﬂexive, whether it is transitive, whether is it antisymmetric. Draw the directed graph that represents each relation. (a) R = { (1 , 1) , (2 , 2) , (2 , 3) , (3 , 2) , (3 , 3) , (3 , 4) , (4 , 3) , (4 , 4) , (5 , 5) } . (b) S = { (1 , 1) , (1 , 2) , (1 , 4) , (1 , 5) , (2 , 2) , (2 , 4) , (2 , 5) , (3 , 3) , (3 , 4) , (3 , 5) , (4 , 4) , (4 , 5) , (5 , 5) } . (c) T = { (1 , 3) , (1 , 5) , (2 , 4) , (3 , 1) , (3 , 5) , (4 , 2) , (5 , 1) , (5 , 3) } . (d) U = { (1 , 1) , (1 , 3) , (1 , 5) , (2 , 2) , (2 , 4) , (3 , 1) , (3 , 3) , (3 , 5) , (4 , 2) , (4 , 4) , (5 , 1) , (5 , 3) , (5 , 5) } . Hints : To solve such questions, ﬁrst draw the directed graphs (arrow diagrams) and then decide whether they are reﬂexive, or transitive or antisymmetric. (a) R = { (1 , 1) , (2 , 2) , (2 , 3) , (3 , 2) , (3 , 3) , (3 , 4) , (4 , 3) , (4 , 4) , (5 , 5) } . r r r r r 2 3 5 1 4 - ± - ± Each of element has a loop. Please add these loops when you read it. R is reﬂexive, symmetric, but not transitive ((2 , 3) , (3 , 4) are in R but (2 , 4) is not), not anti-symmetric (since both (2 , 3) and (3 , 2) are in R ). (b) S = { (1 , 1) , (1 , 2) , (1 , 4) , (1 , 5) , (2 , 2) , (2 , 4) , (2 , 5) , (3 , 3) , (3 , 4) , (3 , 5) , (4 , 4) , (4 , 5) , (5 , 5) } 1

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r r r r 2 4 5 1 3 - @ @ @ @ @ @ @R ? - ± ± ± ± ± ± ±² 6 ³ @ @ @ @ @ @ @I S is reﬂexive, transitive and antisymmetric. (c) T = { (1 , 3) , (1 , 5) , (2 , 4) , (3 , 1) , (3 , 5) , (4 , 2) , (5 , 1) , (5 , 3) } . r r r r r 3 4 5 1 2 - ? ³ ± ± ± ± ± ± ±´ 6 ± ± ± ± ± ± ±² - ³ No loops in this arrow diagram. T is transitive, but not reﬂexive and not antisymmetric. (d) U = { (1 , 1) , (1 , 3) , (1 , 5) , (2 , 2) , (2 , 4) , (3 , 1) , (3 , 3) , (3 , 5) , (4 , 2) , (4 , 4) , (5 , 1) , (5 , 3) , (5 , 5) } . r r r r r 3 4 5 1 2 - ? ³ ± ± ± ± ± ± ±´ 6 ± ± ± ± ± ± ±² - ³ Each element has a loop. Please add them when you read it. U is transitive, symmetric and reﬂexive, but not anti-symmetric. 2. (a) Which (if any) of the relations in question 1 are partial orders on the set A ? In each such case draw the Hasse diagram of the partial order. (b) Which (if any) of the relations in question 1 are equivalence re- lations? In each such case, list the equivalence classes into which the relation partitions A . Hints
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MAS111_09_assignment_10_hints - MAS 111 FOUNDATION OF...

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