Properties of the Given System:

Linearity (P): A system is linear if it satisfies the superposition property. In other words, if an input x1[n] produces an output y1[n] and an input x2[n] produces an output y2[n], then for any constants a and b, the input ax1[n] + bx2[n] should produce the output ay1[n] + by2[n].

Time Invariance (Q): A system is time-invariant if a time shift in the input signal results in the same time shift in the output signal. Mathematically, if an input x[n] produces an output y[n], then an input x[n - k] should produce an output y[n - k], where k is a constant.

Causality (R): A system is causal if the output at any given time depends only on the current and past values of the input. In other words, the output should not depend on future values of the input signal.

Stability (S): A system is stable if bounded inputs produce bounded outputs. In other words, if the input signal is limited in magnitude, the output signal should also be limited in magnitude.

Analyzing the Given Relationship:

The given relationship is Q.y[n] = nx[n]. Let's analyze this relationship to determine the properties of the system.

Property Q: Time Invariance
- The output y[n] is obtained by multiplying the input signal x[n] with n.
- If we shift the input signal by a constant k (x[n - k]), the output signal will also be shifted by the same constant k (y[n - k]).
- Hence, the given system satisfies the time invariance property.

Property R: Causality
- The output y[n] is directly dependent on the current value of the input signal x[n] multiplied by n.
- There is no dependency on future values of the input signal.
- Hence, the given system satisfies the causality property.

Property P: Linearity
- To determine linearity, we need to check if the system satisfies the superposition property.
- Let's consider two inputs x1[n] and x2[n] that produce outputs y1[n] and y2[n] respectively.
- For input x1[n], the output is y1[n] = nx1[n].
- For input x2[n], the output is y2[n] = nx2[n].
- Now, let's consider the input ax1[n] + bx2[n], where a and b are constants.
- The output for this input is Q.y[n] = n(ax1[n] + bx2[n]) = anx1[n] + bnx2[n] = ay1[n] + by2[n].
- Hence, the given system satisfies the linearity property.

Property S: Stability
- To determine stability, we need to analyze if bounded inputs produce bounded outputs.
- Since the input signal x[n] is not specified, we cannot determine if it is bounded or not.
- Therefore, we cannot definitively determine the stability of the system based on the given relationship.

Conclusion:
Based on the analysis above, the given system