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Discrete Mathematics

Set Operations

Set Operations: Level 1 Challenges


What is \(|A^{C}\cup B^{C}|\)?


\(|S|\) represents the cardinality of the set \(S\) and \(S^C\) represents the complement of the set \(S\).

True or False?

\[\left|X^C \cup Y^C \cup Z^C\right| = \left|(X \cap Y \cap Z)^C\right|\]

If X and Y are two sets, X \(∩\) \((Y∪X)^c\) is equal to

Details and assumptions:
\(\rightarrow\) \(∩\) represents intersection.
\(\rightarrow\) \(∪\) represents union.
\(\rightarrow\) \(X^c\) represents compliment of set X.

In my fridge's shelves there are 10 condiments that are good for breakfasts, 8 that are good for lunch/dinner, and 12 that are good for dessert.

Of all of the condiments, exactly half of those that are good for breakfast are also good for lunch/dinner. And every condiment is either good for desert or good for both lunch/dinner and breakfast.

Given all of this, what is the difference between the maximum number of condiments that I might have in my fridge and minimum number of condiments that I might have in my fridge?

If \(A\) and \(B\) are subsets of \(U=\{1, 2, 3, 4, 5, 6, 7, 8, 9\}\) such that \[ { A }^{ c }\cap { B }^{ { c } }=\{1, 9\}, A\cap B =\{2\}, { A }^{ c }\cap B=\{4, 6, 8\},\] what is the set \(A?\)


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