Chapter 1 — Sets
Collect, describe, compare, combine — the language of sets that every later chapter speaks: domains, solution sets, intervals and events.
Open any page of mathematics written in the last hundred years and you will find sets. The domain of a function is a set; the solutions of an equation form a set; an inequality such as $3x-5\lt7$ is solved by an interval, which is a set; in probability, "getting an even number on a die" is the set $\{2,4,6\}$. Sets are less a topic than a language — the grammar in which the rest of Class 11 and Class 12 mathematics is written. This chapter teaches that language carefully, because a small confusion here ($\in$ or $\subset$? is $0$ a natural number? is $\{0\}$ empty?) causes errors chapters later.
The ideas are everyday ones made exact. A search filter on a shopping app ("phones under ₹15,000 with at least 4.2 stars") describes a set by a property; the results page lists it. A playing XI is a subset of a squad. A survey of who watches cricket and who watches football is a picture of two overlapping sets, and counting the people who watch "either" or "both" is the first real counting problem you will meet. By the end of the chapter you will draw such pictures — Venn diagrams — and calculate with unions, intersections, differences and complements, and with the laws that link them, such as De Morgan's laws.
- decide whether a collection is well-defined, and use $\in$, $\notin$ and the standard sets $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$ correctly;
- write a set in roster form and in set-builder form, and convert between the two;
- recognise empty, finite, infinite and equal sets, and find $n(A)$ by counting distinct elements;
- test whether $A\subset B$, list the subsets of a small set, and write subsets of $\mathbb{R}$ as intervals such as $[a,b)$;
- draw Venn diagrams in a universal set $U$, form $A\cup B$, $A\cap B$, $A-B$ and $A'$, and use the algebraic laws of sets, including De Morgan's laws.
Sections in this chapter
Sets and Their Representations
Well-defined collections, roster and set-builder forms, empty, finite, infinite and equal sets.
Subsets, Intervals and the Universal Set
$A\subset B$, listing subsets, the number systems, intervals on the real line, and the universal set.
Venn Diagrams and Operations on Sets
Union, intersection, difference and complement, their laws, and De Morgan's laws.