Question: $\varepsilon = \{\text{integers } x : 1 \le x \le 12\}$, $A = \{\text{even numbers}\}$, $B = \{\text{factors of } 20\}$. List the elements of $A$, $B$ and $A \cap B$, and describe how the Venn diagram is filled.

Step 1. The universal set fences everything in: only the integers $1$ to $12$ exist for this question.

Step 2. List each set within $\varepsilon$:

($20$ is a factor of itself, but $20 \notin \varepsilon$, so it is excluded.)

Step 3. Intersection — elements in both:

Step 4. Venn placement, working from the middle outwards: $2, 4, 10$ in the overlap; then $6, 8, 12$ in $A$ only; $1, 5$ in $B$ only; the leftovers $3, 7, 9, 11$ outside both circles (but inside the $\varepsilon$ box).

Check. Count: $3 + 3 + 2 + 4 = 12$ elements — everything in $\varepsilon$ placed exactly once. ✓

⚠ Watch out: Always re-read the universal set before listing — dropping $20$ from $B$ is the point of the question. Fill Venn diagrams from the intersection outwards; writing all of $A$ into the "A only" region double-counts the overlap.

Question: In an estate, $C = \{\text{families owning a car}\}$ and $M = \{\text{families owning a motorcycle}\}$. In the Venn diagram, $x, y, z, k$ count the regions: $C$ only, $C \cap M$, $M$ only, and neither. Given $n(\varepsilon) = 350$, $n(C) = 200$, $n(M) = 120$: (a) find the smallest possible $k$; (b) the largest possible $y$; (c) $x$, if $z = 50$.

Step 1. (a) $k$ (neither) is smallest when the circles overlap as little as possible, spreading $C$ and $M$ out:

Step 2. (b) $y$ (both) is largest when one set swallows the other entirely — capped by the smaller set:

Step 3. (c) With $z = 50$: the motorcycle circle splits as $n(M) = y + z$:

Step 4. Then the car circle: $n(C) = x + y$:

Check. Total: $130 + 70 + 50 + k = 350 \Rightarrow k = 100$ — a consistent, non-negative filling. ✓

⚠ Watch out: $n(C) = 200$ counts the whole circle ($x + y$), not the "$C$ only" region — the single most common misread. Extreme-value parts are geometry: minimum overlap pushes circles apart, maximum overlap nests them.