Discrete Mathematics
# Set Operations

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 \(B = \emptyset\).

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

Options:

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

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

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

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

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

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

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.

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

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