Discrete Mathematics

# Sets - Operations on Multiple sets

Consider a ground set $$U$$ and let $$A,B$$ and $$C$$ be subsets of $$U$$ such that \begin{align} & \lvert U \rvert =110, \lvert A \rvert =44, \lvert B \rvert =27, \\ & \lvert C \rvert =47, \lvert A \cup B \cup C \rvert=90. \end{align} If no element belongs to exactly two of the three subsets, what is the number of elements in the complement of $$A \cap B \cap C?$$

Consider a ground set $$U$$ and let $$X,Y$$ and $$Z$$ be subsets of $$U$$ such that \begin{align} \lvert U \rvert & =140, \lvert X \rvert =40, \lvert Y \rvert =35, \lvert Z \rvert=53, \\ \lvert X \cap Y \rvert &=7, \lvert Y \cap Z \rvert=10, \lvert Z \cap X \rvert=15, \\ & \lvert X \cap Y \cap Z \rvert=3. \end{align} What is the number of elements in the complement of $$X \cup Y \cup Z?$$

Given a universal set $$U$$ and subsets $$A, B$$ and $$C$$ of $$U$$, which of the following is equivalent to the set $\left\{ A\cap (A^{ c }\cup B) \right\} \cup \left\{ B\cap (B\cup C) \right\}?$

$$100$$ students took a quiz with the three problems Easy, Medium, and Challenge. $$52$$ solved Easy, $$35$$ students solved Medium, and $$32$$ students solved Challenge. Also, $$37$$ students solved exactly two out of the three problems and $$10$$ students solved all three problems. How many students solved none of the three problems?

Suppose $$A$$, $$B$$ and $$C$$ are three sets such that $A \cap B=\emptyset, \lvert A \rvert =9, \lvert B \rvert=9,$ $\lvert C \rvert=6, \lvert A \cup C \rvert =14, \lvert B \cup C \rvert=10.$ What is the number of elements in the set $A \cup B \cup C?$

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