Final answer:
The system has the property of linearity, but does not have the properties of time invariance, causality, and stability. The correct option is 1) Linearity.
Step-by-step explanation:
The system property of linearity indicates that a system follows the principle of superposition, which means that if inputs are scaled and added together, the outputs will also scale and add together accordingly.
In this case, the system y[n] = tx[n] = x[-n] is linear because if we input two scaled and added signals, the resulting output will also be scaled and added. For example, if we input a signal x1[n] + x2[n], the output will be t(x1[n] + x2[n]) = tx1[n] + tx2[n]. Therefore, the system has the property of linearity.
The system property of time invariance indicates that a system's response does not change over time. In this case, the system y[n] = tx[n] = x[-n] is not time-invariant because the output depends on the value of n, which represents time.
When we shift the input signal x[n] by a certain time delay, the output signal will also be shifted by the same time delay. Therefore, the system does not have the property of time invariance.
The system property of causality indicates that a system's output depends only on past and present inputs. In this case, the system y[n] = tx[n] = x[-n] is not causal because the output y[n] depends on the negative values of n, which represent future inputs. Therefore, the system does not have the property of causality.
The system property of stability indicates that a system does not produce unbounded outputs for bounded inputs. In this case, the system y[n] = tx[n] = x[-n] is unstable because it produces unbounded outputs for bounded inputs. For example, if we input a bounded signal x[n], the output will be unbounded since it depends on the value of t. Therefore, the system does not have the property of stability.