Question

5.49 Determine which of the following are functions with the given domains $X$ and codomains $Y$. For those that are functions, determine which are injective, surjective, or bijective. (a) $X = Y = mathbb{Z}$ and $f = {(x, -3x) mid x in X}$. (b) $X = [-1, 0]$, $Y = [-1, 1]$, and $f = {(x, y) in X imes Y mid x^2 + y^2 = 1}$. (c) $X = [-1, 1]$, $Y = [-1, 0]$, and $f = {(x, y) in X imes Y mid x^2 + y^2 = 1}$. (d) $X = [-1, 0]$, $Y = [-1, 0]$, and $f = {(x, y) in X imes Y mid x^2 + y^2 = 1}$. (e) $X = Y = mathbb{N}$ and $f = {(m, n + 1) in mathbb{N} imes mathbb{N} mid m ext{ has exactly } n ext{ distinct prime factors}}$. For example, $(80, 3) in f$ because $80 = 2^4 cdot 5$ has two distinct prime factors. (f) $Y = {1, 2, 3, 4}$, $X = mathcal{P}(Y)$, and $f = {(x, y) in X imes Y mid ext{the smallest element of } x ext{ is } y}$. For example, $({1, 2, 4}, 1) in f$.

          5.49 Determine which of the following are functions with the given domains $X$ and codomains $Y$. For those that are functions,
determine which are injective, surjective, or bijective.
(a) $X = Y = mathbb{Z}$ and $f = {(x, -3x) mid x in X}$.
(b) $X = [-1, 0]$, $Y = [-1, 1]$, and $f = {(x, y) in X 	imes Y mid x^2 + y^2 = 1}$.
(c) $X = [-1, 1]$, $Y = [-1, 0]$, and $f = {(x, y) in X 	imes Y mid x^2 + y^2 = 1}$.
(d) $X = [-1, 0]$, $Y = [-1, 0]$, and $f = {(x, y) in X 	imes Y mid x^2 + y^2 = 1}$.
(e) $X = Y = mathbb{N}$ and
$f = {(m, n + 1) in mathbb{N} 	imes mathbb{N} mid m 	ext{ has exactly } n 	ext{ distinct prime factors}}$.
For example, $(80, 3) in f$ because $80 = 2^4 cdot 5$ has two distinct prime factors.
(f) $Y = {1, 2, 3, 4}$, $X = mathcal{P}(Y)$, and
$f = {(x, y) in X 	imes Y mid 	ext{the smallest element of } x 	ext{ is } y}$.
For example, $({1, 2, 4}, 1) in f$.

5.49 Determine which of the following are functions with the given domains X and codomains Y. For those that are functions,
determine which are injective, surjective, or bijective.
(a) X = Y = mathbbZ and f = (x, -3x) mid x in X.
(b) X = [-1, 0], Y = [-1, 1], and f = (x, y) in X imes Y mid x^2 + y^2 = 1.
(c) X = [-1, 1], Y = [-1, 0], and f = (x, y) in X imes Y mid x^2 + y^2 = 1.
(d) X = [-1, 0], Y = [-1, 0], and f = (x, y) in X imes Y mid x^2 + y^2 = 1.
(e) X = Y = mathbbN and
f = (m, n + 1) in mathbbN imes mathbbN mid m ext has exactly n ext distinct prime factors.
For example, (80, 3) in f because 80 = 2^4 cdot 5 has two distinct prime factors.
(f) Y = 1, 2, 3, 4, X = mathcalP(Y), and
f = (x, y) in X imes Y mid extthe smallest element of x ext is y.
For example, (1, 2, 4, 1) in f.

Added by Steve G.

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