Set Theory • Topic 2 of 3

Complement

The complement A' is everything in the universal set U that is NOT in A, so n(A') = n(U) − n(A). Complements turn awkward "not" conditions into easy subtractions: the number who do NOT like cricket is just the total minus the cricket lovers. The two De Morgan’s laws tie complements to union and intersection: (A∪B)' = A'∩B' and (A∩B)' = A'∪B'. In words, "not in either" equals "outside both", and "not in both" equals "missing from at least one". The first law is the CAT workhorse: "neither A nor B" is exactly (A∪B)', so n(neither) = n(U) − n(A∪B). A clean tactic on survey questions is to compute n(A∪B∪C) first, then subtract from the total to get the "none" region in one move, rather than chasing each negative condition separately.

✅ Solved examples

1. U has 50 elements and n(A) = 19. Find n(A’).
n(A') = n(U) − n(A) = 50 − 19 = 31.
2. In a town of 500, 320 own a two-wheeler and 210 own a car; 90 own both. How many own neither?
Own at least one = 320 + 210 − 90 = 440. Neither = (A∪B)' = 500 − 440 = 60.
3. Verify De Morgan: U = {1..10}, A = {1,2,3,4}, B = {3,4,5,6}. Show (A∪B)’ = A’∩B’.
A∪B = {1,2,3,4,5,6} ⇒ (A∪B)' = {7,8,9,10}. A' = {5,6,7,8,9,10}, B' = {1,2,7,8,9,10}; A'∩B' = {7,8,9,10}. Equal.
4. Of 240 people, 150 like tea, the rest do not. Of the tea-lovers 60 also like coffee, and 30 like coffee only. How many like neither drink?
Coffee total = 60 + 30 = 90; tea = 150 with 60 in both. At least one = 150 + 90 − 60 = 180. Neither = 240 − 180 = 60.

✏️ Practice — try these, take hints as needed

1. U has 80 elements, n(B) = 53. Find n(B’).
n(B') = n(U) − n(B).
80 − 53.
Just subtract.
27
2. In a class of 120, 70 passed Maths and 80 passed English; 50 passed both. How many failed both subjects?
At least one pass = 70 + 80 − 50.
= 100.
Neither = (A∪B)' = 120 − 100.
20
3. If n(U) = 100, n(A∪B) = 72, how many are in neither A nor B?
Neither = (A∪B)'.
n(U) − n(A∪B).
100 − 72.
28
4. A’ has 18 elements and U has 45. Find n(A).
n(A) = n(U) − n(A').
45 − 18.
Complement of the complement.
27
5. U = {1..12}, A = {2,4,6,8}, B = {1,2,3,4}. Find n((A∩B)’).
A∩B = {2,4}.
(A∩B)' = U minus {2,4}.
12 − 2.
10

📝 Topic test — 8 questions

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