Q.If $U = \{1, 2, 3, \dots, 10\}$ and $A = \{1, 3, 5, 7, 9\}$, find the complement $A'$.

The complement of $A$ is everything in the universal set left over once $A$'s own elements are removed.

Removing the odd numbers 1, 3, 5, 7, 9 from $U$ leaves exactly the even numbers.

Step 1 — Recall the definition. $A' = U - A$, the set of elements in $U$ that are not in $A$.

Step 2 — List $U$ and remove $A$'s elements. $U = \{1,2,3,4,5,6,7,8,9,10\}$. Removing $1,3,5,7,9$ leaves $2,4,6,8,10$.

Step 3 — Cross-check by counting. $n(U) = 10$, $n(A) = 5$, so $n(A') = 10 - 5 = 5$, matching the 5 elements listed.

As a shortcut, once $n(U)$ and $n(A)$ are known, $n(A') = n(U) - n(A)$ can be found first as a check before listing $A'$ element by element.

Forgetting to fix the universal set $U$ first, or including elements outside $U$ — the complement is always taken relative to a stated $U$, never in the abstract.