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I sort of understand a and b, however, please give some ideas how to prove them all. I am new to proof.

(a) Define a "parenthesizing" of a union of n sets $\bigcup_{i=1}^{n}X{i}.$ Similarly, define a "parenthesizing" of a sum of n numbers of $\sum_{i=1}^n a_{i}$

(b) Prove that any two parenthesizing of the intersection $\bigcap_{i=1}^{n}X_{i}$ yield the same result.

(c) How many ways are there to parenthesize the union of 4 sets $A \cup B \cup C \cup D$?

(d) Try to derive a formula or some other way to count the number os ways to parenthesize the union of n sets $\bigcup_{i=1}^{n}X_{i}$.

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    For 3 sets: $A,B,C$ we can write this as: \begin{align*} A \cup B \cup C &= (A \cup B) \cup C \\ &= A \cup (B \cup C) \\ &= B \cup (A \cup C) \end{align*} Now try the rest of the question in a similar fashion – 2017-01-14

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I only answer (c), (d)

(c): $$(AB)(CD), ((AB)C)D, A((BC)D), (A(BC))D, A(B(CD))$$ $5$ ways

There is another way of thinking. By simple check, there are $2$ ways for parenthesizing $ABC$, i.e. $(AB)C, A(BC)$. In case of parenthesizing $ABCD$, first parenthesize two elements, i.e. $(AB)CD, A(BC)D, AB(CD)$. Each case, there are $2$ ways for parenthesizing. But $(AB)(CD)$ is overlapped, so exclude one case. The answer is $3\cdot 2-1=5$ Following this way, the algorithm can be generalized to answer (d). But there is much elegant answer for (d)

(d): Catalan number $C_n$ is number of parenthesizing $n+1$ number of sets.