What are the definitions of memoryless, linear, and stable properties?

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The discussion focuses on the definitions of memoryless, linear, and stable properties in the context of mathematical systems. Memoryless properties indicate that the future state of a system depends only on its current state, not on past states. Linear properties refer to systems that adhere to the principles of superposition and homogeneity. Stability pertains to the behavior of a system in response to external disturbances, ensuring that it returns to equilibrium after perturbations. These definitions are crucial for understanding system dynamics in various fields such as control theory and signal processing.

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Hi, I need to prove/disprove the following statements. I am facing lots of difficulty though. Intuitively my guess is that they are all true, but I cannot begin to think how I would prove them.

I have attached the questions and appreciate any help.

Thanks.
 

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You need to apply the definitions of these properties. What are the definitions of memoryless, linear, stable?
 

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