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CAT 2025 Lesson : Set Theory - Concepts & Cheatsheet

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Note: The video for this module contains a summary of all the concepts covered in this lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

   8. Cheatsheet

11) n(A∪B)n(\text{A} \cup \text{B}) =n(A)+n(B)−n(A∩B)= n(\text{A}) + n(\text{B}) - n(\text{A} \cap \text{B})

22) n(A∪B∪C)n(\text{A} \cup \text{B} \cup \text{C}) =n(A)+n(B)+n(c)−n(A∩B)−n(B∩C)−n(A∩C)= n(\text{A}) + n(\text{B}) + n(\text{c}) - n(\text{A} \cap \text{B}) - n(\text{B} \cap \text{C}) - n(\text{A} \cap \text{C}) + + n(A∩B∩C) n(\text{A} \cap \text{B} \cap \text{C})

33) Where P, Q and R are the number of elements in exactly 1, exactly 2 and all 3 sets respectively,

(a) n\bm{n}(A  ∪  \ \bm{\cup} \ B  ∪ \ \bm{\cup} \ C) == P + Q + R
(b) n\bm{n}(A) ++ n\bm{n}(B) ++ n\bm{n}(C) == P + 2Q + 3R

44) Where P, Q, R and S are the number of elements in exactly 1, exactly 2, exactly 3 and all 4 sets respectively,
(a) n\bm{n}(A  ∪  \ \bm{\cup} \ B  ∪ \ \bm{\cup} \ C  ∪ \ \bm{\cup} \ D) == P + Q + R + S
(b) n\bm{n}(A) ++ n\bm{n}(B) ++ n\bm{n}(C) ++ n\bm{n}(D) == P + 2Q + 3R + 4S

55) When number of elements in each set (i.e., nn(A), nn(B), ...) and that of universal set (i.e., nn(U)) are given, then
11) Maximum number of elements in all sets == Minimum of (n(\bm{(n(}A),n(\bm{), n(}B),...)\bm{), ...)}
22) Minimum number of elements in all sets == U−n(A)‾\bold{U} - \bm{n}\bold{(\overline{A)}} +\bold{+} n(B)‾\bm{n}\bold{(\overline{B)}} +...)\bold{+ ...)}

(Note: nn(A‾\overline{A}) == 100−n100 - n(A), nn(B‾\overline{B}) =100−n= 100 - n(B),...)

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