exercises_3 - Analysis: Exercises 3 1. [in F.T.A.] (a) Let...

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Unformatted text preview: 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 R2 where (a, b) ∼ (c, d) if and only if a2 + b2 = c2 + d2 . Prove that this is an equivalence relation, and describe the equivalence classes. 2. Let f : R → R defined by f (x) = x3 − 1. Let a ∈ R. Find (i) f (1); (ii) f (a); (iii) f (a + 1); (iv) f (a − 1); (v) 2f (2a). 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) = x2 +1 x. 4. Let f : R − {0} → R be a function satisfying 1 (∀x ∈ R − {0}) f (x) + 2f ( ) = x. x Find f (x) for every x ∈ R − {0}. 5. Let f : Z2 → Z2 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 (∀x1 ∈ X )(∀x2 ∈ X )[(f (x1 ) = f (x2 )) ⇒ (x1 = x2 )]. State the negation of the above definition. (b) Let f : X → Y . Then f is called a surjection if (∀y ∈ Y )(∃x ∈ X )[f (x) = y ]. State the negation of the above definition. 7. For each of the following functions, decide (giving a proof) whether it is injective, whether it is surjective, and whether it is bijective. (a) f : N → N given by f (n) = 2n + 1. (b) g : R → R, where g (x) = x if x < 0 and g (x) = (x − 1)3 if x (c) h : (0, 1) → R+ given by h(x) = (1 − x)/x. 0. 2 ...
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This note was uploaded on 02/25/2010 for the course MATHEMATIC 11007 taught by Professor Spyros during the Winter '10 term at Bristol Community College.

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exercises_3 - Analysis: Exercises 3 1. [in F.T.A.] (a) Let...

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