4023f10ps4

4023f10ps4 - R is an equivalence relation and write down...

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Homework #4 Due: October 4, 2010 1. For each of the following relations R on the set X determine whether R is re±exive, symmetric, antisymmetric, or transitive. (a) X = Z aRa if and only if a b + 1; (b) X = Z aRb if and only if a + b is even; (c) X = Z aRb if and only if a + b is odd; (d) X = Z aRb if and only if a and b are relatively prime; (e) X = R aRb if and only if a ¡ b is an integer; (f) X = R aRb if and only if a 2 b 2 . You may record your answers (yes/no) in the following table: Relation Re±exive Symmetric Antisymmetric Transitive (a) (b) (c) (d) (e) (f) 2. Problem 2.4, Page 38 in Lax. 3. Let X be the set f a; b; c; d; e g and let R be the relation on X with the matrix M R = 2 6 6 6 6 4 1 0 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 0 0 0 0 1 3 7 7 7 7 5 : Verify that R is a partial order on X and draw the Hasse diagram of R . 4. Let X be the set f 1 ; 2 ; 3 ; 4 g and let R = f (1 ; 1) ; (1 ; 2) ; (2 ; 1) ; (2 ; 2) ; (3 ; 3) ; (3 ; 4) ; (4 ; 3) ; (4 ; 4) g : Show that
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Unformatted text preview: R is an equivalence relation and write down its equivalence classes. 5. Let S be the set f 1 ; 2 ; 3 ; 4 g and let X be the set S S . Dene a relation R on X by the rule ( a; b ) R ( c; d ) if and only if a + b = c + d . Show that R is an equivalence relation and list the equivalence classes. 6. Recall that if f : X ! Y is a function, then an equivalence relation R f is dened on X by the rule x 1 R f x 2 if and only if f ( x 1 ) = f ( x 2 ). For the function f : R 2 ! R by f ( a; b ) = a + b , determine the equivalence classes of each of the following elements of R 2 : (a) (1 ; 1), (b) (1 ; 1), (c) ( 2 ; 3). Math 4023 1...
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