Probability
# 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, **$X \cap (Y \cup X)^c$** is equal to

**Details and assumptions**:

$\rightarrow$ **$\cap$** represents intersection.

$\rightarrow$ **$\cup$** 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?