Is the Discrete Time System x[n] → y[n] = x[-n] Time Invariant?

[ElijahRockers]

Homework Statement

I am supposed to determine wether or not the discrete time system

x[n] \rightarrow y[n] = x[-n]

is time invariant or not.

The Attempt at a Solution

Let x_d[n] = x[n-n_0]

y_d[n] = x_d[-n] = x[-(n-n_0)] = x[-n+n_0]

y[n-n_0] = x[-(n-n_0)] = x[-n+n_0]

Since y_d[n] = y[n-n_0], shouldn't this prove time invariance?

The book says the answer is that it is not time invariant...

From the more qualitative definition, a time invariant system is one for which the behavior does not change depending on when it is evaluated...
Now, I see that for -ve values of n, the system looks ahead, and for +ve values of n the system looks behind. Would this be considered time variant because of this? If so, how do I go about showing that mathematically?

 

[ElijahRockers]

I have just run into a similar problem, where y[n] = Even{x[n-1]}.

When I try shifting the input, then shifting the output and comparing them, the expressions are equal, but the book is telling me the system is not time invariant.