Let i, j, ke Za and w, x, y e Z, with i, j, k all different from eachother and w, x, y all different from each other. When afunction f: Za → Zz is chosen at random, what is Pr[f(k) = y |%3Df(i) = w and f(j) = x]?%3DExpress your answer numerically using plain text withoutspaces. Use / and ^ and ! to represent division,exponentiation and factorial. Use parenthesis around a termonly if it is necessary. Examples: 1/(x-1), (x-3)/10, x!, (x-1)!,x^2, 2^(x-1), etc.

Answered: Let i, j, k e Za and w, x, y e Zb with…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 11E: For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions...

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Question

Let i, j, ke Za and w, x, y e Z, with i, j, k all different from each
other and w, x, y all different from each other. When a
function f: Za → Zz is chosen at random, what is Pr[f(k) = y |
%3D
f(i) = w and f(j) = x]?
%3D
Express your answer numerically using plain text without
spaces. Use / and ^ and ! to represent division,
exponentiation and factorial. Use parenthesis around a term
only if it is necessary. Examples: 1/(x-1), (x-3)/10, x!, (x-1)!,
x^2, 2^(x-1), etc.

Expert Solution

Step 1

Let the number of elements in the domain be equal to m, and the number of elements in Z_b be equal to n.

The total number of possible functions from Z_a to Z_b is equal to n^m.

The number of functions that have f(x)=y are:

Match k to y ,we still have m-1 elements in Z_a to match, each element in Z_a can be matched to any arbitrary elemt in Z_b. There are n^m-1 possibilities.

Match i to w, j to x, we still have m-2 elements in Z_a to match, each element in Z_a can be matched to any arbitrary elemt in Z_b. There are n^m-2 possibilities.

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