Can you please help me to solve and understand this question with proper explanation and mathematical steps?

Let the probability space (0, F, P), where ? = {1, 2,3}, F = 2", and P satisfies P((1} ) = 0.5 and P({2} ) = 0.1. Consider the random variable X : 0 - R defined by X (w) = 2w+3. (a) Find X-1([1,5)) and X-'([2, 4] U {7}). (b) Derive the probability distribution of X as the induced measure PX-. Be sure to define it for any Borel set BE B. (c) What is the support of X? (d) Derive the cumulative distribution function (cdf) of X. Be sure to define it for any TER. (e) Is X discrete or continuous? Justify