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# ps07 - MAT 300 Mathematical Structures Problem Set 7...

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MAT 300, Mathematical Structures Dr. L. Mantini Problem Set 7 Solutions Spring 2010 5.1 : 2, 4, 6, 14; 5.2 : 2, 4, 5, 6, 8, 9, 11, 17, 18 5.1.2 Is each relation f a function? (a) We have A = { a,b,c,d } , B = { x,y,z } , and f ( a ) = y , f ( b ) = x , and f ( c ) = y . NO , this is not a function since no value of f ( d ) is given. (b) Let W be the set of all English words and A the set of all letters of the alphabet. Deﬁne f : W A and g : W A by f = { ( w,a ) | letter a occurs in word w } and g = { ( w,a ) | letter a is the ﬁrst letter of word w } . NO , f is not a function since a word with more than one letter has more than one value. YES , g is a function since every English word has a unique ﬁrst letter. (c) Let P = { J,M,S,F } , and arrange the letters in P around a circle. Then deﬁne R = { ( p,q ) P × P | p is immediately to the right of q } . YES , R : P P is a function since each letter in P has a unique nearest neighbor to the right. 5.1.4 (3 points) Determine the requested values of the given functions. (a) Let N be the set of all countries and C the set of all cities. Let H : N C be deﬁned by H ( n ) = capital of n . What is H (Italy)? Solution : H (Italy) = Rome. (b) Let A = { 1 , 2 , 3 } and B = P ( A ), the set of all subsets of A . Let F : B B be deﬁned by F ( X ) = A \ X . What is F ( { 1 , 3 } )? Solution : F ( { 1 , 3 } ) = A \ { 1 , 3 } = { 2 } . (c) Let f : R R × R where f ( x ) = ( x +1 ,x - 1). What is f (2)? Solution : f (2) = (3 , 1). 5.1.6 Let f and g be the functions from R to R given by f ( x ) = 1 / ( x 2 +2) and g ( x ) = 2 x - 1. Find f g and g f . Solution : ( f g )( x ) = f (2 x - 1) = 1 (2 x - 1) 2 + 2 = 1 4 x 2 - 4 x + 3 , and ( g f )( x ) = g ( f ( x )) = 2 f ( x ) - 1 = 2 x 2 + 2 - 1 = - x 2 x 2 + 2 . The next exercise uses the deﬁnition of a constant function , which has two equivalent versions. We may use either version. Deﬁnition 1: Let A and B be sets. A function f : A B is constant if there exists a b B such that f ( x ) = b for all x A . Deﬁnition 2: Let A and B be sets. A function f : A B is constant if there exists x 1 A such that f ( x ) = f ( x 1 ) for all x A . 5.1.14 (6 points) Suppose A is a nonempty set and f : A A .

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(a) Suppose there is some a A such that f ( x ) = a for all x A . Prove that for all g : A A , f g = f . Proof
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ps07 - MAT 300 Mathematical Structures Problem Set 7...

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