This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ...
View
Full
Document
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.
 Winter '10
 Spyros

Click to edit the document details