Set Notation

Set Notation: Level 2 Challenges


How many sets A exist for which {1,2}A{1,2,3,4,5}\{1,2\} \subseteq A \subseteq \{1,2,3,4,5\} ?

Solve for XX.

As a true connoisseur of capsaicin, you can taste the presence of every single one your XX chili species distinctly in any chili sauce. So there are a LOT of chili sauces in your kitchen. In fact, your kitchen is basically filled with the 81928192 tiny, semi-explosive chili sauce bottles you own. Why so many? -- because you have every kind of chili sauce that uses any combination of any number of your chilies (including one INSANE chili sauce that has all XX of the kinds of pepper you love in it!!!! -- whenever you use it, there's a 20% chance of spontaneous combustion of the food it's put on.) You also have exactly one chililess chili sauce among the 81928192, just to complete the set... but anyone who uses it is a dodo bird.

Calculate |P(P(A)) \mathcal{P}(\mathcal{P}(A))| if A is the Empty Set and P(X)\mathcal{P}(X) means "the power set of set XX."

In how many subsets of {1,2,3,4,5} \{1,2,3,4,5\} is the sum of the largest element and the smallest element equal to 6?

X is a subset of Y and Y is a subset of Z. Which of the following statements must be true?

(A)  \ \ If 1 is in Y, then 1 is in X.
(B)  \ \ If 2 is in Z, then 2 is in X.
(C)  \ \ If 3 is in Z, then 3 is in Y.
(D)  \ \ If 4 is in X, then 4 is in Z.
(E)  \ \ None of the above


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