- #1

ArcanaNoir

- 779

- 4

## Homework Statement

Show that [itex] (\mathbb{Z} _4 , \oplus ) \approx ( \mathbb{Z} ^{0}_{5} [/itex] [itex] , \odot ) [/itex]

Meaning, they are isomorphic. The 0 means the zero is deleted from the set. We are using circle plus and circle dot because we are not allowed to think of the operations as addition and multiplication yet. My trouble is with defining the operation [itex] \theta [/itex].

## Homework Equations

I know [itex] \theta [/itex] must be operation preserving, that is, [itex] \theta (x \oplus y) = \theta (x) \odot \theta (y) [/itex]

## The Attempt at a Solution

I tried defining [itex] \theta [/itex] as [itex] \theta ([x])= [x+1] [/itex]

so, [itex] \theta ([0]) = [1] [/itex] up to [itex] \theta ([3]) = [4] [/itex]

And now, [itex] \theta ([x] \oplus [y]) = \theta ([x \oplus y]) = [x \oplus y +1] [/itex]

Did I make the wrong steps there? I'm not sure how to arrive at [itex] \theta ([x]) \odot \theta ([y]) [/itex]

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