Sets

A set is a Many that allows itself to be thought of as a One. — Georg Cantor

1.1 Introduction

A set is a well-defined collection of objects. Objects in the set are elements (or members). If $a$ belongs to $A$, we write $a\in A$; otherwise $a\notin A$.

Sets are written in roster form $\{1,2,3\}$ or set-builder form $\{x:x\text{ is a prime},\ x<10\}$.

1.2 Types of Sets

• Empty set $\phi$: no elements
• Finite / infinite: finitely many or infinitely many elements
• Equal sets: same elements ($A=B$)
• Subset: $A\subset B$ if every element of $A$ is in $B$. $A\subseteq B$ allows $A=B$
• Proper subset: $A\subset B$ and $A\mathrel{\htmlClass{course-neq}{\char"2260}} B$
• Universal set $U$: the ambient set for a discussion

1.3 Venn Diagrams and Operations

Union: $A\cup B=\{x:x\in A\text{ or }x\in B\}$

Intersection: $A\cap B=\{x:x\in A\text{ and }x\in B\}$

Difference: $A-B=\{x:x\in A,\ x\notin B\}$

Complement: $A'=U-A$

If $A\cap B=\phi$, the sets are disjoint.

Laws (selected)

$A\cup B=B\cup A,\qquad A\cap B=B\cap A$

$(A\cup B)'=A'\cap B',\qquad (A\cap B)'=A'\cup B'\quad\text{(De Morgan)}$

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$

for finite sets.

Chapter Summary

• Set: well-defined collection; $\in$, $\notin$
• Roster and set-builder forms
• $\phi$, finite/infinite, subset, universal set
• $\cup$, $\cap$, $-$, complement; De Morgan; counting formula for union

Exercises (NCERT)

• Exercise 1.1 onwards — representations, subsets, operations
• Miscellaneous Exercise