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exercises_3 - Analysis Exercises 3 1[in F.T.A(a Let f X Y...

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Analysis: Exercises 3 1. [in F.T.A.] (a) Let f : X Y be a function. Define the relation R on X 2 by R = { ( x, y ) X × X | f ( x ) = f ( y ) } . Prove that R is an equivalence relation. (b) Let be the relation on R 2 where ( a, b ) ( c, d ) if and only if a 2 + b 2 = c 2 + d 2 . Prove that this is an equivalence relation, and describe the equivalence classes. 2. Let f : R R defined by f ( x ) = x 3 - 1. Let a R . Find (i) f (1); (ii) f ( a ); (iii) f ( a + 1); (iv) f ( a - 1); (v) 2 f (2 a ). 3. Determine the range of the following functions. (a) f : Z Z , f ( x ) = x + 1; (b) f : Z Z , f ( x ) = | x | + 1; (c) f : R - { 0 } → R , f ( x ) = x 2 +1 x . 4. Let f : R - { 0 } → R be a function satisfying ( x R - { 0 } ) f ( x ) + 2 f ( 1 x ) = x. Find f ( x ) for every x R - { 0 } . 5. Let f : Z 2 Z 2 defined by f (( m, n )) = ( m, 0). Determine (a) f ( A ), where A = { (0 , n ) | n Z } ; (b) f ( B ), where B = { ( n, n ) | n Z } ; (c) f - 1 ( C ), where C = { ( m, 0) | m N } ; (d) f - 1 ( D ), where D = { ( m, 0) | m Z } . 6. (a) Let f : X Y . Then f is called an injection if ( x 1 X
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