The number systems as a chain of containments, and intervals as pieces of R

What to assume they know

• Containment, proper containment, and why the empty set is inside everything — containment, and containment failing
• Listing the members versus stating the property they share — set-builder form
• The standing symbols N, Z, Q, R from §1.2
• That a rational number can be written as one integer over another with a non-zero denominator
• That the square root of 2 and π are not rational
• The real number line, and reading a point's position on it

What they should be able to do

• State the printed containment relations among the number systems, including the one that is a non-containment
• Write the rationals in set-builder form and say why the denominator condition cannot be dropped
• Recognise members of the rationals that are not written as fractions, and give the fraction that shows they qualify
• Define the irrationals by exclusion and give members of that set
• Translate between an interval and its set-builder form, in both directions
• Read the four bounded bracket shapes off a number-line picture and back
• Explain what a round bracket at an infinite end is doing, and why a square one never appears there in this section
• Compute the length of a bounded interval and say what it does not tell you about how many points the interval holds

Where it usually goes wrong

• "The irrationals are one more link in the chain." They are not: T is what is left of R after Q is taken out, so T and Q sit side by side inside R and neither contains the other. The printed non-containment involving the naturals is the chapter's own warning about this.
• "Every rational is written as a fraction." –5 is rational; it is a whole number on the page and a quotient the moment you need it to be.
• "The denominator condition is a technicality." Drop it and the description stops describing numbers at all. It is part of the defining property.
• "A square bracket means the interval is bigger." It means one more point is in — the endpoint. The stretch between the ends is identical.
• "An interval with a short length has few points." Every interval, however short, holds endlessly many points; the chapter says so directly under Fig 1.1. Length measures reach, not count.
• "Length tells you which interval you have." The closed interval from 6 to 12 and the one from 6 to 12 missing its left end both have length 6.
• "You can close the infinite end." Nothing sits at infinity to be included, and every unbounded end printed in this section carries a round bracket.
• "[a, b] makes sense when a equals b." The section opens by taking the first end strictly below the second, so the degenerate case is outside what is being defined here.

Questions to check understanding

• Write a described set of reals as an interval, and an interval in set-builder form
• Give the length of a stated interval
• Decide containment between two intervals and justify it at the endpoints
• State which of the number-system relations hold and disprove one that does not by naming a single member
• Express a number that does not look rational as a quotient of integers

Examples worth working on the board

Values marked verified are an added derivation from the chapter's own data; the chapter prints no answers to its exercises.

• The four systems as printed (§1.6.1, pp. 10–11). N is given as 1, 2, 3, 4, 5 and onwards; Z runs endlessly both ways through 0; Q is given in set-builder form as the quotient of two integers with a non-zero divisor; T is defined as the reals left over once Q is removed.
• Members of Q that do not look rational (§1.6.1, p. 11): –5, five sevenths, three and a half, and negative eleven thirds. Verified: –5 qualifies as –5 over 1, and three and a half as 7 over 2 — the chapter gives these two rewritings on the page, and they are the whole point of the item.
• Members of T (§1.6.1, p. 11): the square root of 2, the square root of 5, and π.
• The printed relations (§1.6.1, p. 11): the naturals inside the integers inside the rationals; the rationals inside the reals; the irrationals inside the reals; and the naturals not inside the irrationals. That last one is the item to teach — it is a non-containment printed alongside four containments, and one member disproves it, since 1 is a natural number and is rational.
• The interval definitions (§1.6.2, pp. 11–12). Two reals are taken with the first strictly below the second. Four bracket shapes follow: neither end included; both ends included; the first included and the second not; the second included and the first not.
• Fig 1.1 (p. 11) — four number-line strips in a row, captioned with the four bracket forms in this order: neither end, both ends, first end only, second end only. On each strip the two ends are marked a and b beneath, the stretch between them is drawn in a contrasting colour, an included end is a filled dot and an excluded end is a hollow circle. Read off the printed page: the first strip has two hollow ends, the second two filled, the third a filled left and a hollow right, the fourth a hollow left and a filled right. This is the figure the explanation cannot do without.
• The unbounded forms (§1.6.2, p. 11): the non-negative reals as an interval from 0 with no upper end; the negative reals as an interval up to 0 with no lower end; and the whole real line written with both ends unbounded. In all three the infinite end carries a round bracket, which is what the page prints.
• A containment between intervals (§1.6.2, p. 11): the open interval from –3 to 5 sits inside the closed interval from –7 to 9. Verified: –7 ≤ –3 and 5 ≤ 9, so every point of the first is a point of the second; the containment is proper, since –7 itself is in the second and not the first.
• Two translations (§1.6.2, p. 11). The reals above –5 and up to and including 7 become an interval; and the interval from –3 up to but excluding 5 becomes a set-builder description. Both are printed worked.
• Exercise 1.3 Q5 (p. 13) — four descriptions to be written as intervals: the reals above –4 and at most 6; the reals strictly between –12 and –10; the reals from 0 up to but excluding 7; the reals from 3 to 4 inclusive. Verified: the four answers are, in order, half-open with the right end included; open; half-open with the left end included; and closed. Their lengths are 10, 2, 7 and 1.
• Exercise 1.3 Q6 (p. 13) — four intervals to be turned back into set-builder descriptions: open from –3 to 0; closed from 6 to 12; from 6 to 12 with only the right end included; from –23 to 5 with only the left end included. Verified: the lengths are 3, 6, 6 and 28 — note that the second and third have equal length and are different sets, which is the cleanest possible demonstration that length does not identify an interval.

Figures to have open

• A redraw of Fig 1.1 (p. 11): four horizontal number lines, each with its two ends marked, the segment between them emphasised, filled dots for included ends and hollow circles for excluded ones, and the bracket form captioned above each. This is the chapter's own figure and the argument of sections 7 and 8 depends on it; redraw as a clean schematic rather than reproducing the printed art.
• A nested-region diagram of the number systems with the irrationals shown as the complement of the rationals inside the reals. Standard schematic; the chapter states the relations in symbols and draws no picture of them.
• A single number line carrying two same-length intervals that differ at one end, for section 10. Standard schematic.

Where this sits in the book

• NCERT Class 11 Mathematics, Chapter 1 "Sets", §1.6.1 Subsets of set of real numbers, pp. 10–11, and §1.6.2 Intervals as subsets of R, pp. 11–12
• Fig 1.1, p. 11
• The definition of interval length, p. 12
• Exercise 1.3, questions 5 and 6, p. 13
• Backward pointer inside the same chapter: the standing symbols for the number systems are set up in §1.2, p. 2