**7.10 Let \( f: S \to T \).**a) Show that \( f \) is one-to-one if and only if there exists a function \( g: T \to S \) such that \( g \circ f = i_S \).b) Show that \( f \) is onto if and only if there exists a function \( g: T \to S \) such that \( f \circ g = i_T \).c) Show that \( f \) is one-to-one and onto if and only if there exists a function \( g: T \to S \) such that \( g \circ f = i_S \) and \( f \circ g = i_T \).

Answered: Image /qna-images/answer/046d2dd5-ec5d-43e4-94ec-249c635be114.jpg

Answered: 7.10 Let f: S-T. a) Show that f is…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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1*) Let X = {1,2, 3}, Y = {2,3, 4}, and Z = {1,2}. (a) Define a function f: Z → X that is one-to-one but not onto. (b) Define a function g: X → Z that is onto but not one-to-one. (c) State whether there exists a function h: X →Y that is one-to-one but not onto.
Let A be a nonempty set. Prove that if there is an injective function f :(0,1) -> A, then A is uncountable.
3. Let I(r) numbers such that if r # y then g(x) # g(y). Prove that there is a function f such that fog= I. = z be the compositive identity function. Suppose g is a function with domain of all real
Let fít) = 1,2+1,1,1)||, What is the minimum value of this function of t ? Answer the same question for the functiong defined by g(t) = |(1,2 +1,7– )||.
1. Let A = {1,2,3}, B = B = {4,5,6}, The functions f : A → C and g: B → C are defined by f(1) = 7, f(2)= 9, f(3) = 7, Does there exist a function h: A → B such that go h = f? Does there exist a function k : B → A such that fo k = g? Justify your answers. {4,5,6}, C = {7, 8, 9, 10}. g(4) = 10, g(5) = 7, g(6) = 9.
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.]
Which of the following function f: R X R → R is not onto? 1 2x - 3 2 3x3 - x 3 x2-2x 4 f(x) = x2- 5x + 5
= 2x. Let f: RR be given by f(x) Let g: Z → Z be given by g(x) = 2x. Select the true statements. Of and g have the same range g is one-to-one Og is onto Of is one-to-one Of is onto
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)) =

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