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Set Operations

Familiarize yourself with the foundational notation, tools and concepts for the operations that are applied to 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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