Homework 9 Solutions W11

Homework 9 Solutions W11 - EECS 203: Homework 9 Solutions...

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EECS 203: Homework 9 Solutions Section 8.1 1. (E) 4 a) antisymmetric, transitive b) reflexive, symmetric, transitive c) reflexive, symmetric, transitive d) reflexive, symmetric 2. (E) 10 Only the relation in part (a) is irreflexive. 3. (E) 20 An asymmetric relation R must also be antisymmetric because there are no ( a,b ) R such that ( b,a ) R . This vacuously satisfies the definition of antisymmetric. An antisymmetric relation may not be asymmetric. For example, consider a relation R on the set { a } such that R = { ( a,a ) } . This is antisymmetric but not asymmetric. 4. (E) 24 a) R - 1 = { ( a,b ) | a > b } b) R = { ( a,b ) | a b } 5. (M) 46ab a) There are only two relations on a set with one element, and both are transitive. b) There are 13 transitive relations on a set with two elements, which can easily be found by enumerating them. Alternatively, the following reasoning can be used. Observe that a relation on two elements is always transitive when it does not contain both ( a,b ) and ( b,a ) . There are 2 2 - 1 = 3 ways to select ( a,b ) and ( b,a ) without choosing both at once, and there are 2 2 = 4 ways to select the self-loops ( a,a ) and ( b,b ) . This is 3 · 4 = 12 transitive relations that do not contain both ( a,b ) and ( b,a ) . There is only one transitive relation that contains
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Homework 9 Solutions W11 - EECS 203: Homework 9 Solutions...

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