Sets — set language, notation and Venn diagrams

Reading $\cap$ as "or" and $\cup$ as "and" (or simply swapping the symbols), so the wrong region is selected — e.g. giving the overlap when $A \cup B$ was asked for.

Picking Venn-diagram regions by eye rather than shading the set described and reading off only the shaded parts — this misfires for $A'$, $A \cap B$ and $A \cup B$ alike.

$A'$ means everything in the universal set $\xi$ that is not in $A$ — leaving out elements of $\xi$, or counting elements outside $\xi$, gives the wrong complement and the wrong $n(A')$.

$n(A \cap B)$ is the count of elements in the overlap — a single number. $A \cap B$ is the set — the list of those elements. If a question asks for $n(\dots)$, give a number; if it asks for the set, list the elements.

$A \cap B'$ means "in $A$ and not in $B$" — the part of $A$ outside the overlap ($A$ only), not the overlap itself. The overlap is $A \cap B$; do not confuse the two regions.

For "given the element is in $B$", the sample space shrinks to $B$. Divide by $n(B)$, not by the universal total $n(\xi)$: $P(A \mid B) = \dfrac{n(A \cap B)}{n(B)}$.