Discrete Mathematics

Set Operations

Set Operations: Level 2 Challenges


If \(A\) and \(B\) are subsets of \(U=\{1,2,3,4,5\}\) such that \[A \cup B = U, A\cap B =\emptyset, \] how many possible choices are there for the ordered pair \((A,B)\)?

Details and assumptions

It is possible that \(A = \emptyset\) or \(B = \emptyset\).

Let \(A\) and \(B\) be two sets and \(U\) be a universal set such that \(|U|=700\), \(|A|=200\), \(|B|=300\) and \(|A\cap B|=100\). Find \(|A^{C}\cap B^{C}|\).

Which of the following sequence of set operations counts the number of elements that are in exactly one of \(A\), \(B\) or \(C\)?"


  1. \(|A\cap B|+|B\cap C|+|C\cap A|-3(|A\cap B\cap C|)\)

  2. \(|A|+|B|+|C|-2(|A\cap B|+|B\cap C|+|C\cap A|)+3|A\cap B\cap C|\)

  3. \(|A\cap B|+|B\cap C|+|C\cap A|-2|A\cap B\cap C|\)

  4. \(|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|\)

  5. \(|(A\cap B\cap C)\cup(A\cap B^{C}\cap C^{C})\cup(A^{C}\cap B\cap C^{C})\cup(A^{C}\cap B^{C}\cap C)|\)

  6. |\(A\cup B\cup C\) |

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.

Person "A" says the truth 60% of the time, and person "B" does so 90% of the time.

In what percentage of cases are they likely to contradict each other in stating the same fact?


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