6. If X and Y are finite sets, prove that |X'| = XY, Here, XY is the set of all functions from Y to X. (Hint: Write Y = {y1,y2, . , Ym} and X = {x1,x2,. , Xn}. A function ƒ : Y → X is uniquely determined as soon as f(y1), f(y2), ..., f(ym) are specified. How many choices are there for f(y1), for ƒ(y2), etc.)
6. If X and Y are finite sets, prove that |X'| = XY, Here, XY is the set of all functions from Y to X. (Hint: Write Y = {y1,y2, . , Ym} and X = {x1,x2,. , Xn}. A function ƒ : Y → X is uniquely determined as soon as f(y1), f(y2), ..., f(ym) are specified. How many choices are there for f(y1), for ƒ(y2), etc.)
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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divisible by p; for any i with 1 < i < n.)
5. Prove by induction that 3 divides 7" – 4 for every n E N.
6. If X and Y are finite sets, prove that |XY| = |X|Y. Here, XY is the set of all functions
from Y to X.
Write Y =
(Hint:
uniquely determined as soon as f(y1), f(y2), ... , f (ym) are specified. How many choices are
there for f(y1), for f(y2), etc.)
{Y1, Y2, . . . , Ym} and X =
{x1, x2, . . . , xn}. A function f : Y → X is
7. Let X be any set and let Y = "27X = {x : X → {0, 1}}, that is, Y is the set of all functions
from X to {0, 1}. Show that the map F : Y → P(X) defined below is a bijection:
F : X+ {x € X | x(x) = 1} where xE27X
(We discussed this function along with an example in class on 08/12.)
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