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# problems12-sol - MATH S-104 Lecture 12 In-class Problem...

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Unformatted text preview: MATH S-104 Lecture 12 In-class Problem Solutions July 30, 2009 Problem 1 * : Which of these relations on the set of all functions from Z to Z are equivalence relations? Determine the properties of an equivalence relation that the others lack. a) {( f,g ) f ( 1 ) = g ( 1 )} i. reflexive: a function always equals itself ii. symmetric: if f ( ) = g ( ) , then g ( ) = f ( ) iii. transitive: if f ( ) = g ( ) and g ( ) = h ( ) , then f ( ) = h ( ) b) {( f,g ) f ( ) = g ( ) or f ( 1 ) = g ( 1 )} i. reflexive: yes ii. symmetric: yes iii. transitive: no. What if f ( ) = g ( ) and g ( 1 ) = h ( 1 ) . It could be the case that f ( ) ≠ h ( ) and f ( 1 ) ≠ h ( 1 ) . c) {( f,g ) f ( x )- g ( x ) = 1for all x ∈ Z } i. reflexive: no. f ( x )- f ( x ) = ii. symmetric: no. f ( x )- g ( x ) = 1 means that g ( x )- f ( x ) =- 1 iii. transitive: no. If f ( x )- g ( x ) = 1 and g ( x )- h ( x ) = 1, then f ( x )- h ( x ) = 2....
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problems12-sol - MATH S-104 Lecture 12 In-class Problem...

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