Final answer:
The discrete-time system where y[n] = x[-n] is memoryless and linear but not time invariant or causal; it is stable assuming x[n] is bounded.
Step-by-step explanation:
To determine whether the properties listed hold for the discrete-time system where the system output y[n] is x[-n], we must evaluate each property one by one.
Memoryless: The system is memoryless because the output y[n] at any time n depends only on the input at time -n, and not on any other past or future inputs.
Time invariant: The system is not time invariant. A shift in time for x[n] would not produce a proportional shift in the output y[n], since y[n] would be affected by the reversal in time index.
Linear: The system is linear since the operation does not change if we scale the input or sum multiple inputs.
Causal: The system is not causal because its output depends on future inputs (for example, y[-1] depends on x[1]).
Stable: The system is stable if for a bounded input, the output is also bounded. Assuming x[n] is bounded, y[n] would be bounded as well, thus the system is stable.