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Set Theory

Set Theory

MODULES

Set Notations
Types of Sets
Set Operations & Venn Diagram
2 Set Venn Diagrams
3 Set Venn Diagrams
4 Set Venn Diagrams
Maximum and Minimum
Past Questions

CONCEPTS & CHEATSHEET

Concept Revision Video

SPEED CONCEPTS

Set Theory - 1
-/10

PRACTICE

Set Theory : Level 1
Set Theory : Level 2
Set Theory : Level 3
ALL MODULES

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

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

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

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

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

444) 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}n(A  ∪  \ \bm{\cup} \  ∪  B  ∪ \ \bm{\cup} \  ∪  C  ∪ \ \bm{\cup} \  ∪  D) === P + Q + R + S
(b) n\bm{n}n(A) +++ n\bm{n}n(B) +++ n\bm{n}n(C) +++ n\bm{n}n(D) === P + 2Q + 3R + 4S

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

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

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