Chapter 2.2 — Exercise 2.1 — Roster and Set Builder

Five Collections, Tested

Each collection below is checked against the same question: is there a definite rule for membership, or does it depend on opinion?

• All months of the year beginning with "J" — a set. There are exactly three: January, June, and July, and the rule is objective.
• The ten most talented writers of India — not a set. "Talented" has no fixed rule for deciding who qualifies.
• A team of eleven best cricket batsmen in the world — not a set, for the same reason: "best" isn't something everyone would judge identically.
• All boys in your class — a set. Whether a given student is a boy in your class is a fact you can check.
• All even integers — a set: 2, 4, 6, 8, … — membership (is the number divisible by 2?) is completely unambiguous.

Three of the five collections above hinge on exactly this distinction: "talented," "best," and "beginning with J" all sound similar as English phrases, but only the last one reduces to something checkable. The other two would get a different answer depending on who you asked, which is precisely what disqualifies them — and it's worth noticing that neither "talented" nor "best" is inherently a bad description, only an incomplete one for set-building purposes, since no amount of extra checking resolves the disagreement.

Reading and Writing Membership

With A = {0, 2, 4, 6}, B = {3, 5, 7}, and C = {p, q, r}, filling in ∈ or ∉ becomes a direct check against each set's contents:

Translating Sentences Into Symbols

Question 3 goes the other way, turning plain English into the ∈/∉ notation:

• "The element x does not belong to A" → x ∉ A
• "d is an element of the set B" → d ∈ B
• "1 belongs to the set of natural numbers" → 1 ∈ N
• "8 does not belong to the set of prime numbers P" → 8 ∉ P

True or False, With a Reason

Question 4 asks you to evaluate four membership statements — and every one of them turns out false, which is exactly the kind of question that rewards checking carefully rather than assuming a pattern.

• 5 ∉ the set of prime numbers — false; 5 is itself prime, so it belongs to that set.
• S = {5, 6, 7} implies 8 ∈ S — false; 8 is not listed among S's three elements.
• −5 ∉ W, where W is the whole numbers — true; whole numbers start at 0 and never go negative, so −5 genuinely isn't in W.
• 8/11 ∈ Z, where Z is the integers — false; 8/11 is a fraction, and integers contain no fractions at all.

From Set-Builder to Roster

Question 5 asks for the roster form of three set-builder descriptions — the same translation skill from the chapter introduction, applied to slightly less familiar rules.

• B = {x : x is a natural number smaller than 6} → B = {1, 2, 3, 4, 5}
• C = {x : x is a two-digit natural number whose digits sum to 8} → C = {17, 26, 35, 44, 53, 62, 71, 80}
• D = {x : x is a prime number that divides 60} → D = {2, 3, 5} (60 = 2² × 3 × 5, so only these three primes divide it)

From Roster Back to a Rule

Question 6 reverses the direction — given a plain list of numbers, describe the pattern that generates it.

Spotting the underlying formula — powers of 2, powers of 5, or perfect squares — is what makes this direction harder than roster-to-list conversion: there's usually more than one plausible-looking pattern until you check it against every term. {2, 4, 8, 16, 32}, for instance, could momentarily look like "even numbers" until the fifth term, 32, rules that out — even numbers would also include 6, 10, 12, and every other even value in between, none of which appear in the given list. Checking a candidate rule against every term, not just the first two or three, is what separates a correct set-builder description from a coincidental match.

One Last Roster Round

Question 7 asks for three more roster forms, mixing a numeric range, an algebraic condition, and a word's letters in one set:

• A = {x : x is a natural number greater than 50 but smaller than 100} → A = {51, 52, 53, …, 98, 99}
• B = {x : x is an integer, x² = 4} → B = {−2, 2} (both square to 4)
• D = {x : x is a letter in the word "LOYAL"} → D = {L, O, Y, A} (L appears twice but is listed once)

Matching Descriptions to Lists

The closing question pairs four roster forms with four set-builder descriptions, and the fastest way through it is to translate each roster form into words first, then hunt for the matching rule.

Notice how {1, 2, 3, 6} and {2, 3} are easy to mix up if you only skim the descriptions — both are "divisors of 6," but only one restricts to prime divisors, which is the detail that separates the two matches. Reading a set-builder description all the way through, rather than stopping at the first familiar-looking phrase, is what catches this kind of near-duplicate reliably.

Carrying This Forward

Every skill drilled here — checking whether a collection is well-defined, and converting freely between roster and set-builder form — is assumed throughout the rest of the chapter. Exercise 2.2 builds on it directly, introducing special sets like the empty set and universal set alongside the union, intersection, and difference operations. For the definitions this exercise draws on, revisit the chapter introduction.