Why writing a pair in order carries information a set cannot

What to assume they know

• Set notation from Chapter 1: braces, membership with ∈, the empty set written φ
• That a set is determined by which elements belong to it and by nothing else, so listing the same elements in a different order gives the same set
• Subsets, intersection and union, at the level of naming which elements survive
• Solving a pair of one-step linear equations in one unknown each
• Plotting a point in the coordinate plane from its two coordinates

What they should be able to do

• Explain why a two-element set cannot record which of its elements was chosen first, and produce a concrete pair of situations that the set notation confuses
• Write an ordered pair in the chapter's notation and name its first and second element
• State what the cartesian product of two named non-empty sets contains, and list it in full for small sets
• Decide whether two given ordered pairs are equal, using the coordinatewise test rather than by comparing the elements as a collection
• Solve an equation between two ordered pairs by splitting it into one equation per position, as Example 1 does
• Explain why the product is empty as soon as one of the two factors is empty, and why the other factor's size makes no difference to that
• Read the chapter's grid illustrations (Figs 2.1, 2.2, 2.3) as a picture of a product, identifying which axis supplies first elements and which supplies second

Where it usually goes wrong

• "(a, b) is just another way of writing {a, b}." The braces version is unchanged when you swap the two symbols; the brackets version is not. Show the two side by side on the licence-plate example, where the swap changes which symbol is the state.
• "Two pairs are equal if they contain the same things." They are equal when the openings agree and the closings agree. The set holding m and n and the set holding n and m are the same set; the pair opening with m and the pair opening with n are different pairs.
• "The product of two sets is a set of sets." Its members are pairs, and a pair is a single object with two slots, not a two-element collection.
• "φ has no elements, so crossing with it should leave the other set alone." Multiplication by 1 leaves a number alone; this is not that. Every pair in the product needs a second entry, and there is nothing available to supply one, so nothing survives.
• "Order only matters when the two sets are different." Exercise 2.1 Q4(i) puts the same set on both sides and the four pairs are still four, with two of them distinguished only by order.
• "Fig 2.1 is an arrow diagram." It is a grid of crossings. The chapter's first arrow diagram is Fig 2.4, in §2.3, and arrows carry a different meaning there — they pick out a chosen part of the product rather than displaying all of it.

Questions to check understanding

• Given an equation between two ordered pairs whose entries are expressions, solve for the unknowns
• List the cartesian product of two small named sets in full, and count it
• Decide whether a stated claim about a product is true, and correct it if not — the form Exercise 2.1 Q4 takes
• Given a product listed in roster form, recover the two sets it came from
• Say what the product becomes when one factor is the empty set, with a reason
• Explain, in words, the difference between a two-element set and an ordered pair

Examples worth working on the board

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

• The opening illustration (§2.2, p. 24). A set of two colours, red and blue, and a set of three objects written b, c and s, standing for a bag, a coat and a shirt. The six pairs are printed in full: red with each of b, c, s, then blue with each of b, c, s. Verified: 2 × 3 = 6, and the six are distinct because no two agree in both positions.
• Fig 2.1 (p. 24). Not an arrow diagram. It is a small grid: the two colour names sit along the bottom, the three object letters run up the left side, and a dot marks each of the six crossings. Read from the printed page — the axis lettering is inside the artwork and does not extract.
• Fig 2.2 (p. 25). The same grid shape at 3 × 3: DL, MP and KA along the bottom for Delhi, Madhya Pradesh and Karnataka; 01, 02 and 03 up the left side as licence-plate codes; nine crossing dots. The chapter's restriction is that a code has to open with a state abbreviation. Verified: 3 × 3 = 9 available codes, and the chapter prints all nine pairs.
• The order test the chapter states (p. 25). A code opening with DL and closing with 01 is a different code from one opening with 01 and closing with DL. Both are built from the same two symbols; only the positions differ.
• Fig 2.3 (p. 25). A 2 × 4 grid: two labels a₁ and a₂ along the bottom, four labels b₁ to b₄ up the left side, eight crossing dots. Read from the printed page. The chapter's point about this one is that if the two sets are sets of real numbers, each pair is the position of a point, and the point whose first coordinate is a₁ and second is b₂ is elsewhere on the page from the point whose coordinates are those two numbers exchanged. Verified: 2 × 4 = 8.
• Example 1 (p. 26). The pair whose entries are x + 1 and y − 2 is declared equal to the pair whose entries are 3 and 1. Verified: the equality splits into x + 1 = 3 and y − 2 = 1, giving x = 2 and y = 3.
• Exercise 2.1 Q1 (p. 27). The same shape with fractions: the pair whose entries are x/3 + 1 and y − 2/3 is set equal to the pair whose entries are 5/3 and 1/3. Verified: x/3 = 5/3 − 1 = 2/3 so x = 2; y = 1/3 + 2/3 = 1.
• Exercise 2.1 Q4(i) (p. 27). A true-or-false item worth working: P is given as the set holding m and n, Q as the set holding n and m, and the claim is that the product contains just two pairs. Verified: P and Q are the same set, so the product holds four pairs — m with m, m with n, n with m, n with n. The claim is false, and it is false precisely because a set does not remember the order its members were typed in.
• Exercise 2.1 Q4(iii) (p. 27). A is the set holding 1 and 2, B the set holding 3 and 4, and the question is what A crossed with the intersection of B and φ comes to. Verified: that intersection is φ, and a product with an empty factor is empty, so the answer is φ.

Figures to have open

• The grid of Fig 2.2 redrawn as a schematic: state abbreviations on the horizontal, code numbers on the vertical, a dot at each crossing, and the ability to highlight one dot and show the pair it stands for. This is the chapter's own figure (p. 25) and section 6 depends on it; redraw rather than reproduce.
• A two-panel comparison of a set and an ordered pair under the same swap. Standard schematic.
• A coordinate plane carrying a point and its coordinate-exchanged twin, matching the claim the chapter makes about Fig 2.3 (p. 25). Standard schematic.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 2 "Relations and Functions", §2.1 Introduction, p. 24; §2.2 "Cartesian Products of Sets", pp. 24–26.
• Definition 1 and the Remarks numbered (i) to (iv), pp. 25–26.
• Example 1, p. 26; Example 6, p. 27.
• Exercise 2.1 items 1 and 4, p. 27. (Items 3 and 6 are worked in Counting all the pairs, and why swapping the two sets gives something else, which owns the counting and the recover-the-factors work.)
• Summary, p. 41, for the ordered-pair and cartesian-product entries and for the statement that a product with φ is empty.
• Figs 2.1 (p. 24), 2.2 and 2.3 (p. 25).