Question 4. Let X = {o,0,A, ©} and Y = {a,b, c, d, e}. Determine whether each of the following represent functions. Explain. If the relation is a function, determine the domain, codomain, and таnge. (a) f : X → Y defined via f = {(o,a), (0, 6), (A, c), (O, d)}. (b) д: X —>Ү defined via g 3D {(o, a), (0, b), (A, c), (0, c)}. {(o, a), (0, b), (A, c), (0, d)}. {(0, a), (0, b), (A, c), (0, d), (0, e)}. (c) h : X → Y defined via h = (d) k : X → Y defined via k = (e) l: X → Y defined via l = {(0, e), (I, e), (A, e), (0, e)}. (f) т: X > Y defined viaт — {(o,a), (A, b), (0, c)}. (9) happy : Y → X defined via happy(y) = © for all y E Y. (h) id : X → X defined via id(x) = x for all x € X. One useful representation of functions on finite sets is via bubble diagrams (like in Figure 12.3 of the textbook). To draw a bubble diagram for a function f : X → Y, draw one circle (i.e, a “bubble") for each of X and Y and for each element of each set, put a dot in the corresponding set. Typically, we draw X on the left and Y on the right. Next, draw an arrow from x E X to y E Y if f(x) = y (i.e., (x, y) E f).
Question 4. Let X = {o,0,A, ©} and Y = {a,b, c, d, e}. Determine whether each of the following represent functions. Explain. If the relation is a function, determine the domain, codomain, and таnge. (a) f : X → Y defined via f = {(o,a), (0, 6), (A, c), (O, d)}. (b) д: X —>Ү defined via g 3D {(o, a), (0, b), (A, c), (0, c)}. {(o, a), (0, b), (A, c), (0, d)}. {(0, a), (0, b), (A, c), (0, d), (0, e)}. (c) h : X → Y defined via h = (d) k : X → Y defined via k = (e) l: X → Y defined via l = {(0, e), (I, e), (A, e), (0, e)}. (f) т: X > Y defined viaт — {(o,a), (A, b), (0, c)}. (9) happy : Y → X defined via happy(y) = © for all y E Y. (h) id : X → X defined via id(x) = x for all x € X. One useful representation of functions on finite sets is via bubble diagrams (like in Figure 12.3 of the textbook). To draw a bubble diagram for a function f : X → Y, draw one circle (i.e, a “bubble") for each of X and Y and for each element of each set, put a dot in the corresponding set. Typically, we draw X on the left and Y on the right. Next, draw an arrow from x E X to y E Y if f(x) = y (i.e., (x, y) E f).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Question 4. Let X = {o,0,A, O} and Y = {a, b, c, d, e}. Determine whether each of the following
represent functions. Explain. If the relation is a function, determine the domain, codomain, and
range.
(a) f : X →Y defined via f = {(0, a), (0, b), (A, c), (O, d)}.
(b) g : X → Y defined via g =
{(o, a), (D, b), (A, c), (O, c)}.
(c) h: X → Y defined via h = {(, a), (D, b), (A, c), (0, d)}.
(d) k : X → Y defined via k = {(0, a), (0, b), (A, c), (O, d), (O, e)}.
(e) l: X → Y defined via l = {(0, e), (O, e), (A, e), (O, e)}.
(f) т: X — Y defined via т —
{(o, a), (A, b), (O, c)}.
(g) happy : Y → X defined via happy(y)
= © for all y E Y.
(h) id : X → X defined via id(x) = x for all x € X.
One useful representation of functions on finite sets is via bubble diagrams (like in Figure
12.3 of the textbook). To draw a bubble diagram for a function f : X → Y, draw one circle (i.e,
a “bubble") for each of X and Y and for each element of each set, put a dot in the corresponding
set. Typically, we draw X on the left and Y on the right. Next, draw an arrow from x e X to
УEY if f(*) — у (i.е., (х, у) € f).
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