Overlapping circles that make "how many" questions clear — only, both, neither and all/some/none, with four tiers of practice.
Page 1 · Introduction
Venn diagrams & sets
A Venn diagram uses overlapping circles to show how groups relate — who likes both, who likes only one, who likes neither. It turns confusing "how many" questions into a clear picture.
The overlap in the middle means "in both groups".
The big ideaThe overlap is the key. People in both groups are counted in the middle — and only once.
On the real testWhere this appearsSet and "all/some/none" reasoning is part of the Thinking Skills section (25% of your mark, from 2025).
Page 2 · Reading a diagram
What each region means
Three regions
Left only (12): play soccer but NOT tennis.
Middle (5): play BOTH.
Right only (8): play tennis but NOT soccer.
So the number who play soccer altogether = 12 + 5 = 17 (the whole left circle, including the overlap).
✅"Soccer" means the WHOLE soccer circle — left-only PLUS the middle. Don’t forget the overlap belongs to both circles.
Page 3 · The key distinction
"Only" versus "in total"
The most important skill: telling apart "how many do only X" from "how many do X in total".
The rule
Only X = the X circle MINUS the overlap.
X in total = the X-only part PLUS the overlap.
If 20 like apples and 8 like both apples and bananas: apples-only = 20 − 8 = 12.
💡Watch that little word "only". It tells you to subtract the overlap. Missing it is the #1 mistake on these questions.
Page 4 · Adding up
How many are in at least one group?
To count everyone in either group, add the two groups but subtract the overlap once (or you’d count the middle twice).
At least one = A + B − both
18 play soccer, 12 play tennis, 5 play both. How many play at least one?
18 + 12 − 5 = 25. The overlap (5) was counted in both the 18 and the 12, so subtract it once.
25
✅Why subtract? Because the 5 "both" people are inside the 18 AND the 12. Adding gives them twice; subtracting once fixes it.
Page 5 · The outside
How many are in neither group?
Some people are in no group at all — they sit OUTSIDE both circles. Find them by taking the "at least one" total away from everyone.
Neither = Total − (at least one)
30 students. 18 soccer, 12 tennis, 5 both. How many play neither?
At least one = 18 + 12 − 5 = 25. Neither = 30 − 25 = 5.
5
💡Always ask if there’s an "outside" group. "Neither" people are easy to forget, but the test loves to ask about them.
Page 6 · Logic words
All, some and none
Venn diagrams also handle logic statements:
What they look like
All A are B — circle A sits entirely inside circle B.
Some A are B — the circles overlap.
No A are B — the circles are completely separate.
Careful"All cats are animals" does NOT mean "all animals are cats". The cat circle sits inside the animal circle — direction matters!
Page 7 · Practice
Level 1 · Easy
Warm up
Read the diagram. Tap to check.
1. From the diagram, how many play soccer only?
A 5
B 13
C 18
D 8
Soccer only = the left-only region = 13 (not counting the 5 in the middle).
2. How many play both sports?
A 13
B 8
C 5
D 21
Both = the middle overlap = 5.
3. How many play soccer in total?
A 18
B 13
C 5
D 26
Soccer total = left-only + both = 13 + 5 = 18 (the whole soccer circle).
🎉You’ve got the three regions. Now let’s calculate the tricky "only" and "neither" totals.
Page 8 · Practice
Level 2 · Medium
Only & at least one
20 children like apples, 15 like bananas, and 8 like both.
1. How many like apples ONLY?
A 20
B 12
C 8
D 28
Apples only = 20 − 8 (the both) = 12.
2. How many like bananas ONLY?
A 15
B 8
C 7
D 12
Bananas only = 15 − 8 = 7.
3. How many like at least one of the two fruits?
A 27
B 35
C 43
D 20
At least one = 20 + 15 − 8 = 27. (Adding 20 and 15 counts the 8 twice, so subtract it once.)
💪Notice "only" means subtract the overlap, and "at least one" means add then subtract once. Those two moves solve most set questions.
💡The full method: find "at least one", subtract from the total for "neither", subtract the overlap for "only". Three reliable moves.
Page 10 · Challenge
Level 4 · Challenge
Think smart
At a party of 40 people, 22 drink tea, 18 drink coffee, and 10 drink both.
1. How many drink ONLY ONE of the two drinks (tea or coffee, but not both)?
A 30
B 40
C 20
D 10
Tea only = 22 − 10 = 12. Coffee only = 18 − 10 = 8. Only one drink = 12 + 8 = 20.
2. How many drink neither tea nor coffee?
A 5
B 10
C 0
D 8
At least one = 22 + 18 − 10 = 30. Neither = 40 − 30 = 10.
3. "All the coffee drinkers also drink tea." True or false?
A False — only 10 of the 18 coffee drinkers also drink tea
B True
C Cannot tell
D True, all 18
Only 10 drink both, but there are 18 coffee drinkers — so 8 drink coffee without tea. "All" is false.
🌟You handled "only one of the two", "neither", and an all/some check — the full range of set reasoning. Genuinely impressive.
Complete
Set solved!
You’ve completed the Venn Diagrams & Sets module.
You can now read the three regions, tell "only" from "in total", count "at least one" by subtracting the overlap, find the "neither" group, and handle all/some/none logic. Overlapping-group questions are now a picture you can read.