Suppose the possible values of X are {xi}, the possible values of Y are {yj}, and the poss...

Suppose the possible values of X are {xi}, the possible values of Y are {yj}, and the possible values of X + Y are {zk}. Let Ak denote the set of all pairs of indices (i,j) such that xi + yj = zk ;that is, Ak ={(i,j) : xi + yj = zk}. 

(a) Argue that

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(b) Show that

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(c) Using the formula from part (b), argue that

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(d) Show that

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(e) Prove that