These small finite sets make every count checkable by regions or a tree. “Both” means the intersection; with three sets, a pairwise intersection includes anyone in all three. The same person or option is never counted twice in a union. Contexts are examples, not collected learner data.
A · Library makerspace interests
Out of 30 fictional visitors, 17 chose sketching (S), 13 chose coding (C), and 5 chose both. Thus |S∪C| = 17+13−5 = 25; 5 chose neither. Region check: S only 12, both 5, C only 8, neither 5; 12+5+8+5=30. Do not say all 30 chose an activity.
B · Find the unknown overlap
In a fictional 40-person workshop, 22 chose drawing (D), 18 chose music (M), and 32 chose at least one. Hence |D∩M| = 22+18−32 = 8. Neither = 40−32 = 8. Regions: D only 14, both 8, M only 10, neither 8.
C · Three clubs, with explicit pair meanings
In a fictional 28-person week: art A 12, robotics R 11, choir C 9; |A∩R|=4, |A∩C|=3, |R∩C|=5, and |A∩R∩C|=2. Every pair count includes the two people in all three. The union is 12+11+9−4−3−5+2 = 22; 6 are in none. Exact regions: only A 7, only R 4, only C 3, A&R only 2, A&C only 1, R&C only 3, all three 2. Sum 22. No region is negative, and every set total reconstructs.
D · One session, not two
A fictional afternoon programme lets a participant book one of three distinct studio sessions or one of four distinct outdoor sessions. The two booking categories are mutually exclusive in this choice. Addition gives 3+4=7 possible bookings. It is not 3×4 because a person books exactly one session. If a session appears in both lists, the overlap must first be removed.
E · Two independent stages
A fictional display label has 3 shape choices and 4 caption choices. Every shape works with every caption. One finished label contains one shape and one caption, so multiplication gives 3×4=12 labels. This is not a permutation of the seven items; the stages have different roles.
F · Branch, then multiply within each branch
A fictional community event offers one of two exclusive delivery routes. A visual card has 2 theme choices and 3 layouts: 6 cards. An audio card has 2 narration choices and 2 pacing choices: 4 cards. One chooses visual or audio, never both in this model: 2×3 + 2×2 = 10 routes. The case makes no claim that a particular mode suits a particular kind of learner; these are optional presentation choices.