Each key word in a set question maps to one symbol: “and” and “both” give ∩, “or” gives ∪, and “not” or “only” brings in the complement ′.
This lesson belongs to SPM Mathematics set operations. It uses the notation from reading union, intersection and complement notation and feeds into two-set Venn diagram questions.
One situation, six statements
Here is an original situation. ξ is the set of Form 4 students in a class. Let N = {students in the netball team} and C = {students in the choir}.
| In words | In symbols | Venn region |
|---|---|---|
| In both netball and choir | N ∩ C | The overlap |
| In netball or choir or both | N ∪ C | Both circles entirely |
| In netball but not choir | N ∩ C′ | Netball-only part |
| In choir only | N′ ∩ C | Choir-only part |
| In neither netball nor choir | (N ∪ C)′ | Outside both circles |
| Not in both activities at once | (N ∩ C)′ | Everything except the overlap |
The last two rows look alike and mean different things, which is where most marks are lost.
The mistake that costs marks
Take “neither netball nor choir”. A student splits it into two negatives, writes N′ ∪ C′, and reads it as “not in N, and not in C”.
The symbol ∪ means “or”, so N′ ∪ C′ means “not in N, or not in C (or both)”. That covers everyone except those in both activities, which is the set (N ∩ C)′.
Test with numbers. If 10 students are in N only, 4 in both, 6 in C only and 5 in neither, then (N ∪ C)′ has 5 students, while N′ ∪ C′ has 10 + 6 + 5 = 21.
Reading symbols back into words
Translation must also work in reverse. Given n(N′ ∩ C) = 6, say it as “the number of students in choir but not in netball is 6”.
A useful habit is to shade or sketch the region, then write the notation, then read it back in words. If the reading does not match the original sentence, the notation is wrong.
For example, “in exactly one of the two activities” needs two pieces: (N ∩ C′) ∪ (N′ ∩ C). Here that equals 10 + 6 = 16 students.
Using the translation in a calculation
Suppose a class of 30 has n(N) = 14, n(C) = 12 and n(N ∩ C) = 4. The statement “in neither” is (N ∪ C)′.
First, n(N ∪ C) = 14 + 12 − 4 = 22. Then n((N ∪ C)′) = 30 − 22 = 8 students.
Check yourself
Write each statement in set notation: (a) in choir but not in netball, (b) in netball or choir but not both. Then use the class above to find the number in (b).
Answer
(a) N′ ∩ C.
(b) (N ∩ C′) ∪ (N′ ∩ C), which means netball only together with choir only.
Netball only = 14 − 4 = 10 and choir only = 12 − 4 = 8, so (b) has 18 students.
What to study next
Put the skill to work in the set operations practice set, where each question needs both a diagram and a statement in symbols. For the counting routine behind the numbers, revisit solving three-set Venn diagram questions.
If translation between words and symbols stays difficult, online one-to-one Mathematics tuition gives a teacher time to work through it with you.