Answer to Question #347844 in Discrete Mathematics for bel

Question

Given A =

{1, 2, 3, 4} and B = {x, y, z}. Let R be the following relation from A to B

R = {(1, y), (1, z), (3, y), (4, x), (4, z)}

a. Determine the matrix of the relation

b. Draw the arrow diagram of R

c. Find the inverse relation R^-1of R

d. Determine the domain

and range of R

Answer 1

Sloution:

A={1,2,3,4}; B={x,y,z}; R={(1,y), (1,z), (3,y), (4,x), (4,z)}

a) If R is a relation from A to B and x₁,...,x_m is an ordering of the elements of A and y₁,...,y_n is an ordering of the elements of B, the matrix M_R of R is obtained by defining M_Rij =1 if x_iRy_j and 0 otherwise. So,

M_R=$\begin{vmatrix} 0 &1&1 \\ 0 &0&0\\ 0&1&0\\ 1&0&1 \end{vmatrix}$

b) 1------x,y; 2------; 3-------y; 4---------x,z.

c) We just need to change places of elements: R^-1={(y,1), (z,1), (y,3), (x,4), (z,4)}.

d) R={(1,y), (1,z), (3,y), (4,x), (4,z)}, so: X:1, 1, 3, 4, 4; Domain: {1,3,4}; Y: y, z, x; Range: {y,z,x}.