Why "well-defined" is the whole difference between a set and a heap

What to assume they know

Natural numbers, integers, rational and real numbers as separate number systems
Prime numbers, and factorising a whole number into primes
Solving a quadratic by factorising, e.g. finding both roots of a quadratic whose factors are visible
Reading a two-letter Greek symbol and a struck-through symbol as opposites
The everyday word collection — a card pack, a crowd, a sports team

What they should be able to do

State the condition a collection must satisfy before it may be called a set, and apply it as a decision procedure rather than reciting it
Given a candidate collection, name the test that decides membership and say whether two careful people applying it must agree
Explain why a superlative in a description ("most talented", "most dangerous") usually destroys the test, while a superlative with a fixed rule does not
Use the belongs-to symbol and its negation correctly for a stated element and a stated set
Recall the seven standing symbols for number systems used throughout the book and say which numbers each one collects
Classify a list of nine ordinary collections into sets and non-sets, giving the reason in each case
Distinguish "we cannot list it" from "we cannot decide it", and show that only the second disqualifies a collection

Where it usually goes wrong

"A set has to be a collection of numbers." Three of the chapter's six opening collections are not: rivers, vowels, kinds of triangles. The other three are, so the page puts the two sorts side by side deliberately. The test is about deciding membership, not about what the members are made of.
"Well-defined means you can write the members down." The rivers of India are never listed on the page, and the collection is accepted anyway. Deciding is the requirement; listing is a convenience.
"If it is infinite it is not well-defined." All even integers is accepted in Exercise 1.1. Endlessness is untouched by the test.
"A ranking is fine as long as experts agree." The chapter rejects the renowned-mathematicians collection precisely because agreement is not guaranteed. Contrast it with "the eleven batsmen with the highest career average", which is decidable and would be a set.
"15 is a prime factor of 30 because it divides 30." It divides, but the test names prime factors, and 15 is composite. Students lose this because they read the words and not the test.
"The questions in this chapter is too odd to be a set." The chapter is a fixed printed object, so the test is decidable. Oddness is not a criterion.
"Once a collection is refused, we can still talk about it loosely." Everything the chapter builds afterwards takes a set as input. A refused collection has no union, no complement and no subsets.

Questions to check understanding

Judge each of a printed list of collections and justify every verdict — the form of Exercise 1.1 Q1, where the marks sit in the justification
Insert the belongs-to symbol or its negation between a given object and a given set
Rewrite a rejected collection so that it becomes a set, changing as little of the wording as possible
Explain why a stated collection is not well-defined, naming the part of the description that varies between people
Name the number system each standing symbol stands for

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data; the chapter prints no answer to any exercise.

The six opening collections (§1.2, p. 1). (i) the odd natural numbers below ten, printed out as 1, 3, 5, 7, 9; (ii) India's rivers; (iii) the English alphabet's vowels, printed as a, e, i, o, u; (iv) triangles of various kinds; (v) the prime factors of 210, printed as 2, 3, 5, 7; (vi) the solutions of the quadratic whose left side is x² – 5x + 6, printed as 2 and 3. Verified: 210 = 2 × 3 × 5 × 7, so the four printed primes are exactly the prime factors; and x² – 5x + 6 factorises as (x – 2)(x – 3), giving the two printed roots.
The river test (§1.2, p. 2). Nile is offered as an object the test rejects, Ganga as one it accepts. Use these two as the whole demonstration of section 4 — the collection is never listed, and does not need to be.
The collection that is refused (§1.2, p. 2): the world's five most renowned mathematicians. The chapter's stated reason is that the standard for "most renowned" varies between people.
Two membership readings (§1.2, p. 2). With V the vowels of the English alphabet, a belongs to V and b does not. With P the prime factors of 30, 3 belongs to P and 15 does not. Verified: 30 = 2 × 3 × 5, so P = {2, 3, 5}; 15 divides 30 but is not prime, which is exactly why it is out. This is the best single example in the section — the test is prime factor, not factor.
The seven standing symbols (§1.2, p. 2): N for the natural numbers, Z for the integers, Q for the rationals, R for the reals, and Z⁺, Q⁺, R⁺ for the positive members of the last three of those — the integers, the rationals and the reals. The naturals get no superscripted form, having no negative members to exclude. The chapter says these carry through the whole book.
Exercise 1.1 Q1 (p. 4, running to p. 5) — nine collections to be judged, with justification: (i) the months of a year whose name starts with J; (ii) ten writers of India judged most talented; (iii) an eleven-strong team of the world's best cricket batsmen; (iv) all the boys in your class; (v) all natural numbers below 100; (vi) novels written by Munshi Prem Chand; (vii) all even integers; (viii) the questions in this chapter; (ix) the world's most dangerous animals. Verified: (i), (iv), (v), (vi), (vii) and (viii) carry decidable tests — for (i) the three months are January, June and July; (ii), (iii) and (ix) rest on rankings with no fixed rule and are rejected. Note that (v) and (vii) are accepted while being large or endless: size is not the issue.
Exercise 1.1 Q2 (p. 5). A = {1, 2, 3, 4, 5, 6}; the belongs-to symbol or its negation is to be inserted for 5, 8, 0, 4, 2 and 10. Verified: 5, 4 and 2 belong; 8, 0 and 10 do not.
The chapter's own frame (§1.1, p. 1 and the Historical Note, p. 23). The portrait on the opening page is Georg Cantor, dated 1845–1918; the note at the end records that the unrestricted idea of forming a set of everything was shown in 1902 to break down. Use it in section 6 to say that "well-defined" is an honest working notion, not a finished foundation.

Figures to have open

A decision-gate schematic: an arbitrary object entering, a rule box, and two exits labelled in and out. This is an added device and carries sections 3, 4 and 5. Standard schematic.
A side-by-side of two judges producing two different five-name lists from the same description. Standard schematic; the chapter states the reason but draws nothing.
The portrait of Georg Cantor and the epigraph block on the chapter's opening page (p. 1) are the textbook's own; use a plain caption card instead of reproducing the printed page. The same page also carries a QR code.

Where this sits in the book

NCERT Class 11 Mathematics, Chapter 1 "Sets", §1.1 Introduction and §1.2 Sets and their Representations, pp. 1–2
Exercise 1.1, questions 1 and 2, pp. 4–5
The chapter's closing Historical Note, pp. 22–23, for Cantor's dates and for the 1902 result on the set of all sets