Set Theory

Formula sheet

Everything for this chapter, in one place. $n$ is the total number of people.

Inclusion–exclusion


Two sets	$n(A \cup B) = n(A)+n(B)-n(A \cap B)$
Three sets	$n(A\cup B\cup C) = \sum n(A) - \sum n(A\cap B) + n(A\cap B\cap C)$
Everyone counted once	$n(A\cup B\cup C) = n$ when nobody is left out

Reading the regions

How far an overlap can slide


Largest possible $n(A\cap B)$	$\min\bigl(n(A),\,n(B)\bigr)$
Smallest possible $n(A\cap B)$	$\max\bigl(0,\ n(A)+n(B)-n\bigr)$
Smallest possible triple overlap	$\max\bigl(0,\ n(A)+n(B)+n(C)-2n\bigr)$
Largest possible triple overlap	the smallest of the three set sizes

Symmetric difference


Definition	$A \,\Delta\, B$ = in one or the other, not both
Equivalently	$(A \cup B) - (A \cap B)$
It is associative	so $(P\Delta Q)\Delta(R\Delta S)$ can be taken in any order
An element survives	exactly when it appears in an odd number of the sets

0.2

questions in a typical Quant section

24th

of 25 chapters

≈ 1 of 66 marks a section ride on it.

Asked in7 of 24 papers

20172025

Year by yearper section

Question type8 asked

MCQ 87%Type-in 13%

Type-in answers carry no negative marking.

From every Set Theory question in CAT papers since 2017.