JEE Mathematics Sets Notes Set 02

Sets and Ways of Writing Them

A set is a well-defined collection of distinct objects; the objects are its elements, and \( a\in A \) means that \( a \) belongs to \( A \). 'Well defined' means that for any object we can say without doubt whether it belongs or not.
A set can be written in roster form, by listing the elements once each, for example \( \{2,3,5,7\} \), or in set-builder form \( \{x : P(x)\} \), where \( P(x) \) is the rule satisfied by every element, for example \( \{n^3-1 : n\in\mathbb{N},\ 1\le n\le5\}=\{0,7,26,63,124\} \).

Types of Sets

An empty set \( \varphi \) has no element; a singleton has exactly one; a finite set has a countable number of elements, written \( n(A) \); an infinite set never runs out of elements. Intervals of real numbers are infinite however short they are.
Two sets are equal when they have exactly the same elements; they are equivalent when they merely have the same number of elements. Equal sets are always equivalent but not the other way round.
Take care with a set like \( \{\varphi\} \): it is not empty, it has one element, namely \( \varphi \).

Subsets and the Power Set

\( A\subseteq B \) means every element of \( A \) is in \( B \). Every set is a subset of itself and the empty set is a subset of every set; \( A\subseteq B \) and \( B\subseteq A \) together give \( A=B \).
A set with \( n \) elements has \( 2^n \) subsets and \( 2^n-1 \) proper subsets. The power set \( P(A) \) is the set of all subsets, so \( n(P(A))=2^n \) and the number of subsets of the power set is \( 2^{2^n} \).
Starting from the empty set: \( n(P(\varphi))=1,\ n(P(P(\varphi)))=2,\ n(P(P(P(\varphi))))=4,\ n(P^4(\varphi))=16 \).

Operations and Laws

Union \( A\cup B \) (in A or B), intersection \( A\cap B \) (in both), difference \( A-B=A\cap B' \), complement \( A'=U-A \) and symmetric difference \( A\Delta B=(A-B)\cup(B-A) \) (in exactly one of the two). Sets with empty intersection are disjoint.
Commutative and associative laws hold for union and intersection. Distributive laws: \( A\cap(B\cup C)=(A\cap B)\cup(A\cap C) \) and \( A\cup(B\cap C)=(A\cup B)\cap(A\cup C) \). De Morgan: \( (A\cup B)'=A'\cap B' \) and \( (A\cap B)'=A'\cup B' \).
Useful identities: \( A-(B\cup C)=(A-B)\cap(A-C) \), \( A-(B\cap C)=(A-B)\cup(A-C) \), \( A-(A-B)=A\cap B \), and \( A\cup B=A\cap B \) only if \( A=B \).

Counting Elements

\( n(A\cup B)=n(A)+n(B)-n(A\cap B) \) and \( n(A-B)=n(A)-n(A\cap B) \). For three finite sets,
\[ n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C). \]
Exactly two of the sets: \( \sum n(A\cap B)-3n(A\cap B\cap C) \). Exactly one of the sets: \( \sum n(A)-2\sum n(A\cap B)+3n(A\cap B\cap C) \).
Persons belonging to none of the sets are counted as \( n(U)-n(A\cup B\cup C) \). A Venn diagram filled from the centre outwards makes these counts easy to check.

Example 1. In a group of 80 students, 45 read magazine A, 30 read magazine B and 15 read neither. How many read both?
Answer:
Students reading at least one \( =80-15=65 \).
\( n(A\cap B)=45+30-65=10 \).
10 students read both.

Example 2. Among 1000 consumers, 720 like product P and 450 like product Q. What is the least number who like both?
Answer:
The union cannot exceed 1000, so \( n(P\cap Q)=720+450-n(P\cup Q)\ge1170-1000=170 \).
At least 170 consumers like both.

Example 3. Show that \( A-(A-B)=A\cap B \).
Answer:
\( A-B=A\cap B' \), so \( A-(A-B)=A\cap(A\cap B')'=A\cap(A'\cup B)=(A\cap A')\cup(A\cap B)=A\cap B \).
Hence the identity holds.