Counting all the pairs, and why swapping the two sets gives something else

What to assume they know

• Why writing a pair in order carries information a set cannot — ordered pairs, the cartesian product, and the coordinatewise equality rule
• Intersection and union of sets, and the empty set
• Counting subsets of a finite set: a set with n elements has 2ⁿ of them
• Powers with small whole exponents, and reading 9 as 3 squared
• The coordinate plane, and the idea that a point in space needs three numbers

What they should be able to do

• Derive the size of a product from the construction rather than quoting it, and say why the construction produces no duplicates
• Apply the counting rule to sets whose elements are not listed, given only how many each contains
• Produce a pair of small sets for which the product and its reverse are different sets, and state what quantity they nevertheless share
• Explain, using Remark (iii), why the counting rule says nothing when one factor is infinite, and what the product is like instead
• Write out a threefold product and count it, and name the objects it contains
• Say what the twofold and threefold products of the real numbers with themselves represent geometrically, as Example 5 does
• Check on stated sets whether the product distributes across an intersection and across a union, in the manner of Example 3
• Recover the two factor sets from a product given in roster form, and recover a factor set from the size of the product alone

Where it usually goes wrong

• "A × B ≠ B × A means they are different sizes." They are almost always different sets and always the same size when both factors are finite. Example 2 says both things at once and students hear only the first.
• "× is multiplication, and multiplication commutes." The symbol is borrowed; the operation is not. What survives from arithmetic is the count, not the objects.
• "The counting rule works for any two sets." Remark (iii) is the boundary: if either factor is infinite so is the product, and pq says nothing.
• "The real numbers crossed with themselves is a pair of number lines." It is every point of the plane, one for each pair of coordinates.
• "A threefold product is a product of products." The chapter writes triplets with three slots, not pairs whose opening is itself a pair, and it does not discuss whether the two descriptions amount to the same thing.
• "Example 3 proves the distribution rule." It checks it on one choice of three small sets, and Exercise 2.1 Q7 asks for another check. No general proof appears anywhere in this chapter.
• "You cannot find a set from the size of its product." Exercise 2.1 Q10 gets halfway there, because 9 has only one factorisation as a square, so the size of the set is forced. The size is all the count gives: endlessly many three-element sets have a self-product of nine members, and it takes the two pairs printed in the question to pin down which one is meant.

Questions to check understanding

• Given only the sizes of two sets, state the size of their product, and of the product of one with itself
• Given a product in roster form, recover the two factor sets
• Verify on stated small sets that the product distributes across an intersection or a union
• Count the subsets of a product, given the sizes of the factors
• Decide whether a claim comparing a product with its reverse is about the sets or about their sizes, and answer accordingly
• Deduce a set from the size of its self-product together with a few known pairs

Examples worth working on the board

Values marked verified are worked out here on the chapter's stated data.

• The licence-code count (§2.2, p. 25). Three states and three code numbers. Verified: 3 × 3 = 9, matching the nine pairs the chapter prints and the nine crossing dots of Fig 2.2 read off the printed page.
• Example 2 (p. 26). P holds a, b and c; Q holds the single element r. Verified: the product of P with Q holds three pairs, each opening with a letter of P; the product of Q with P holds three pairs, each opening with r. No pair appears in both, so the two products are different sets, while 3 × 1 and 1 × 3 are both 3. This single example carries the whole argument of sections 4 and 5.
• Example 3 (p. 26). A holds 1, 2, 3; B holds 3, 4; C holds 4, 5, 6. The chapter computes four products. Verified: B ∩ C is the one-element set holding 4, so A crossed with it has 3 pairs. A × B has 3 × 2 = 6 pairs and A × C has 3 × 3 = 9; their intersection is the same 3 pairs. B ∪ C holds 3, 4, 5, 6, so A crossed with it has 3 × 4 = 12 pairs, and the union of A × B with A × C has 6 + 9 − 3 = 12 — the two routes agree, and the inclusion–exclusion count is a useful check the chapter does not perform.
• Example 4 (p. 27). P holds 1 and 2, and the chapter writes out the threefold product in full. Verified: 2 × 2 × 2 = 8 triplets, which is the count printed.
• Example 5 (p. 27). The real numbers crossed with themselves, and crossed with themselves twice. The chapter reads the first as the plane and the second as space.
• Example 6 (p. 27). A product is handed over as four pairs — p with q, p with r, m with q, m with r — and the two factors have to be recovered. Verified: the openings are p and m, the closings are q and r, and 2 × 2 = 4 confirms nothing has been lost.
• Exercise 2.1 Q2 (p. 27). A has 3 elements and B is given as the set holding 3, 4 and 5. Verified: 3 × 3 = 9.
• Exercise 2.1 Q3 (p. 27). G holds 7 and 8; H holds 5, 4 and 2. Verified: each product holds 6 pairs, and they share none.
• Exercise 2.1 Q5 (p. 27). A holds −1 and 1, and the threefold product is asked for. Verified: 2³ = 8 triplets.
• Exercise 2.1 Q6 (p. 27). The product is given as the four pairs a with x, a with y, b with x, b with y. Verified: the factors are the set holding a and b and the set holding x and y.
• Exercise 2.1 Q7 (p. 27). A holds 1, 2; B holds 1, 2, 3, 4; C holds 5, 6; D holds 5, 6, 7, 8. Verified: B ∩ C is empty, so both sides of part (i) are empty and the identity holds trivially here — worth saying, because it is the degenerate case. For part (ii), A × C has 2 × 2 = 4 pairs and every one of them lies in B × D, which has 4 × 4 = 16.
• Exercise 2.1 Q8 (p. 27). A holds 1, 2 and B holds 3, 4. Verified: the product has 4 pairs, so it has 2⁴ = 16 subsets. This count is the hinge on which §2.3 later turns.
• Exercise 2.1 Q9 (p. 27). A has 3 elements, B has 2, and three named pairs are known to lie in the product. Verified: the three openings x, y, z must fill A, and the closings 1 and 2 must fill B.
• Exercise 2.1 Q10 (p. 28). The product of a set with itself has 9 elements, among them the pair opening with −1 and closing with 0, and the pair opening with 0 and closing with 1. Verified: 9 = 3 × 3 forces the set to have 3 elements; the two given pairs contribute −1, 0 and 1, which is already three, so the set is exactly those; the product then has 9 pairs of which 2 are given, leaving 7 to be written out.

Figures to have open

• A fill-in grid whose rows and columns can be relabelled: used for the licence codes, for Example 2 in both directions, and for the counting argument itself. This is the shape of the chapter's own Figs 2.1, 2.2 and 2.3 (pp. 24–25); redraw it as a schematic.
• A plane densely marked with points, and a three-axis frame, to carry Example 5. Standard schematic.
• A two-panel set diagram for Example 3, with B ∩ C and B ∪ C highlighted inside the same picture. Standard schematic.
• Nothing here needs to be reproduced from the printed page.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 2 "Relations and Functions", §2.2 "Cartesian Products of Sets", pp. 24–28.
• Remarks (ii), (iii) and (iv), p. 26.
• Examples 2, 3, 4, 5 and 6, pp. 26–27.
• Exercise 2.1 items 2, 3, 5, 6, 7, 8, 9 and 10, pp. 27–28.
• Summary, p. 41, for the counting rule and for the statement that a product generally differs from its reverse.