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17 Nov 2019
Let X and Y be sets. Recall that a function f from X to Y is a rule which assigns a unique element f(r) E Y to each element r EX. We call X the domain or source of f, we call Y the codomain or target space of f, and we write f : X â Y to indicate that f is a function from X to Y. The image of f is defined to be the set in(f) = {f(x) | x e X} . (Note that the codomain and the image of f are not necessarily equal.) We say that the function f : X-Y is ·surjective or onto if for all y e Y, there exists at least one x E X such that f(x) = y; » injective or one-to-one if for all y im(f) there exists at most one x EX such that f(x)=y; ·bijective if f is both injective and surjective. Problem 1. In parts (a) - (d) below, determine whether the given function is injective, surjective, both, or neither. Justify your answers (a) the function f [0,40,18 defined by f(x)-2 2; (b) the function f : R R defined by g(x) 2r-5; MATH 217 - LINEAR ALGEBRA (c) the function h : R2 â R defined by h(x,y) = 2x2 +5y2; d) the function q : N -> N defined by q(n) HOMEWORK 2, DUE Friday, January 25, 2019, at 1:00am 2 -,if n is odd n/2, if n is even
Let X and Y be sets. Recall that a function f from X to Y is a rule which assigns a unique element f(r) E Y to each element r EX. We call X the domain or source of f, we call Y the codomain or target space of f, and we write f : X â Y to indicate that f is a function from X to Y. The image of f is defined to be the set in(f) = {f(x) | x e X} . (Note that the codomain and the image of f are not necessarily equal.) We say that the function f : X-Y is ·surjective or onto if for all y e Y, there exists at least one x E X such that f(x) = y; » injective or one-to-one if for all y im(f) there exists at most one x EX such that f(x)=y; ·bijective if f is both injective and surjective. Problem 1. In parts (a) - (d) below, determine whether the given function is injective, surjective, both, or neither. Justify your answers (a) the function f [0,40,18 defined by f(x)-2 2; (b) the function f : R R defined by g(x) 2r-5; MATH 217 - LINEAR ALGEBRA (c) the function h : R2 â R defined by h(x,y) = 2x2 +5y2; d) the function q : N -> N defined by q(n) HOMEWORK 2, DUE Friday, January 25, 2019, at 1:00am 2 -,if n is odd n/2, if n is even
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