Problem 1Let f(x)=3. For this problem, we will use the notation cZ to denote the set of multiples of c.For example, 2Z denotes the even integers, and 3Z denotes multiples of 3.(a)Prove that f: Z → Z is not a function (that is, it is not a function if both its domainand codomain are integers.)(b)Prove that f: 2Z → Z is a function (that is, it is a function if its domain is evenintegers and its codomain is all integers.)Note: f is a function from A to B if and only iffor each a € A, f(a) is defined,for each a € A, f(a) does not produce two different outputs, and• for each a € A, f(a) € B.Thus to prove f is a function you would show it has these 3 properties. To show that f is not afunction you would show it does not have at least one of these 3 properties. See our first lectureon functions for examples and as reference on how to structure your proofs in this problem.

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Answered: (a) Prove that f: Z→ Z is not a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Suppose f: A --> B is a function. Define the function F: P(B) --> P(A) (from the powerset of B to the one of A) by setting F(S) = f-1(S) for all S subset of B. Prove or disprove: (1) F is 1-1 if and only if f is onto. (2) F is onto if and only if f is 1-1. [Finite, not infinite.]
Let V be the set of functions f: R → R. For any two functions f, g in V, define the sum f + g to be the function given by (f + g)(x) = f(x) + g(x) for all real numbers. For any real number c and any function f in V, define scalar multiplication cf by (cf)(x) = cf(x) for all real numbers . Answer the following questions as partial verification that V is a vector space. (Addition is commutative:) Let f and g be any vectors in V. Then f(x) + g(x). since adding the real numbers f(x) and g(x) is a commutative operation. (A zero vector exists:) The zero vector in V is the function f given by f(x) = for all. (Additive inverses exist:) The additive inverse of the function f in V is a function g that satisfies f(x) + g(x) = 0 for all real numbers . The additive inverse of f is the function g() for all . me for all real numbers a (Scalar multiplication distributes over vector addition:) If c is any real number and fand g are two vectors in V, then c(f+g)(x) = c(f(x) + g(x)) =
Suppose f: R → R is defined by the property that f(x) = x + x² + x³ for every real number x, and g: R → R has the property that (gof)(x) = = x for every real number a. Then g" (0) = 1/2 1 ○ 1/6

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(a) Prove that f: Z→ Z is not a function (that is, it is not a function if both its domain and codomain are integers.) (b) Prove that f: 2Z → Z is a function (that is, it is a function if its domain is even integers and its codomain is all integers.)