Show that the set of functions (left{f_{1}(x)=x, f_{2}(x)=-x, f_{3}(x)=x^{-1}, f_{4}(x)=-x^{-1} ight}) forms a group under the binary

Show that the set of functions \(\left\{f_{1}(x)=x, f_{2}(x)=-x, f_{3}(x)=x^{-1}, f_{4}(x)=-x^{-1}\right\}\) forms a group under the binary operation of substitution of one function into another. Show that the group is isomorphic to the matrix group of Problem 2.19 .

Data from Problem  2.19

Show that the set of matrices {a, b, c, d\} given by

closes under multiplication and is a representation of the group \(\mathrm{D}_{2}\) in Problem 2.9 .

Transcribed Image Text:

1 a b = (9) = ( 9 ) = ( d ) d = (o C = ( 0 9 )