A relation is nothing more than a chosen part of the product

What to assume they know

• Why writing a pair in order carries information a set cannot — ordered pairs and the cartesian product
• Counting all the pairs, and why swapping the two sets gives something else — the counting rule pq, and the count of subsets 2ⁿ
• Roster and set-builder descriptions of a set, from Chapter 1
• Subsets, and that a set with n elements has 2ⁿ subsets including the empty one and the whole set
• Divisibility, and the primes below 10

What they should be able to do

• State what Definition 2 identifies a relation with, and say what that identification rules out
• Given two small sets and a describing sentence, produce the subset of the product that the sentence picks
• Read a relation off an arrow diagram in roster form, and write the same relation in set-builder form
• Explain why the count of relations from one set to another is a count of subsets, and compute it from the two set sizes
• Compute the number of relations for stated set sizes, and describe the two extreme subsets the count includes
• Explain what changes and what does not when a relation is described as being on a single set rather than from one set to another
• Recover a relation whose data appears only inside a printed figure, as Exercise 2.2 Q4 requires

Where it usually goes wrong

• "A relation is a rule." Definition 2 says subset. Two different sentences can select the same pairs, and the count is 2 to the power pq because a relation is a subset, so counting relations is counting subsets and nothing else. That the count then leaves most subsets of a large product with no short sentence is a consequence of how fast it grows, not the reason for its value — and note that every one of them can still be described the long way, by listing.
• "The arrow diagram is the relation." Remark (ii) calls it a visual representation. Redraw the same four arrows with the ovals swapped left to right and the relation is unchanged.
• "Every element of the first set must appear." Nothing in Definition 2 requires it. Exercise 2.2 Q4's relation happens to use all three, but Example 7 leaves 6 unused as an opening entry, and the definition is untroubled.
• "Every element of the second set must be reached." Fig 2.6 has an element with no arrow at all, and that is permitted.
• "2 to the power pq counts the sentences you could write." It counts the subsets. The gap between the two is the point of the section.
• "A relation on a single set is a different kind of object." The Remark on p. 29 says only that the two sets may be the same one. Example 7 already does it without comment.
• "A relation must be between numbers." The chapter's opening illustration relates letters to names, and §2.1 lists family and classroom relationships alongside mathematical ones.

Questions to check understanding

• Given two sets and a describing condition, write the relation in roster form
• Given an arrow diagram, write the relation in both set-builder and roster form
• Compute the number of relations between two sets of stated sizes
• Decide whether a listed set of pairs is a relation from one named set to another, checking that every pair lies inside the product
• Write in set-builder form a relation supplied only as a picture
• Explain why the same relation can be described by more than one sentence

Examples worth working on the board

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

• The letters-and-names illustration (§2.3, p. 28). P holds the letters a, b and c. Q holds five names: Ali, Bhanu, Binoy, Chandra and Divya. The chosen condition is that the letter opens the name. Verified: the product has 3 × 5 = 15 pairs, which is the count the chapter states, and the condition keeps 4 of them — a with Ali, b with Bhanu, b with Binoy, c with Chandra. Divya survives in neither, because D is not among the three letters; this is the chapter's first hint that an element of the second set can go unused.
• Fig 2.4 (p. 28). Two ovals, three dots in the left one carrying a, b and c, five dots in the right one carrying the five names, and four arrows. Read from the printed page: the lettering is inside the artwork.
• Example 7 (pp. 28–29). A holds 1 through 6, and the relation from A to A is the one where the second entry exceeds the first by one. Verified: the pairs are 1 with 2, 2 with 3, 3 with 4, 4 with 5, 5 with 6 — five of them. Adding one to 6 leaves A, so no sixth pair exists. Fig 2.5 draws both ovals as 1 through 6 with five parallel arrows.
• Example 8 and Fig 2.6 (p. 29). Read from the printed page: the left oval holds 9, 4 and 25; the right oval holds 5, 3, 2, 1, −2, −3 and −5; six arrows are drawn. The relation is that the left entry is the square of the right one. Verified: 9 comes from 3 and from −3, 4 from 2 and from −2, 25 from 5 and from −5 — six pairs, matching the six arrows. The element 1 in the right oval is attached to nothing, since 1 squared is 1 and 1 is not in the left oval.
• The Note on counting (p. 29). If the two sets have p and q elements, the product has pq pairs and the relations number 2 to the power pq.
• Example 9 (p. 29). A holds 1 and 2; B holds 3 and 4. Verified: the product has 4 pairs, so there are 2⁴ = 16 relations. Two of the sixteen are worth showing because they are the ends of the range: the subset that keeps nothing, and the subset that keeps all four.
• Exercise 2.2 Q8 (p. 30). A holds x, y and z; B holds 1 and 2. Verified: the product has 3 × 2 = 6 pairs, so there are 2⁶ = 64 relations.
• Exercise 2.2 Q3 (p. 30). A holds 1, 2, 3, 5; B holds 4, 6, 9; the condition is that the two entries differ by an odd amount. Verified by testing all 3 × 4 = 12 pairs: kept are 1 with 4, 1 with 6, 2 with 9, 3 with 4, 3 with 6, 5 with 4 and 5 with 6 — seven pairs. Discarded are 1 with 9, 2 with 4, 2 with 6, 3 with 9 and 5 with 9. This is the cleanest example in the chapter of a condition selecting an irregular-looking subset.
• Exercise 2.2 Q7 (p. 30). The condition pairs each prime below 10 with its cube. Verified: the primes are 2, 3, 5, 7, so the relation is 2 with 8, 3 with 27, 5 with 125, 7 with 343.
• Exercise 2.2 Q4 and Fig 2.7 (p. 30). Read from the printed page, because nothing about this relation appears in type: the left oval holds 5, 6 and 7, the right oval holds 3, 4 and 5, and three arrows run 5 to 3, 6 to 4, 7 to 5. Verified: every arrow drops the entry by 2, so a set-builder description is that the second entry is two less than the first.

Figures to have open

• Fig 2.4 redrawn as a schematic: two ovals of named dots with arrows that can be switched on one at a time. This is the chapter's own figure (p. 28) and sections 1 and 4 depend on it.
• Fig 2.6 redrawn (p. 29), with all seven right-hand elements present so the unattached one is visible. The seven elements were read off the printed page.
• Fig 2.7 redrawn (p. 30). The relation of Exercise 2.2 Q4 exists nowhere except in this artwork, so the explanation cannot pose that exercise without it.
• A panel able to display the full product as a grid with individual cells switchable, used for the counting argument. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 2 "Relations and Functions", §2.3 "Relations", pp. 28–30.
• Definition 2 and Remarks (i) and (ii), p. 28.
• Examples 7, 8 and 9, and the Note on counting relations, pp. 28–29; the Remark on a relation on one set, p. 29.
• Exercise 2.2 items 3, 4, 7 and 8, p. 30.
• Summary, p. 41, for the entries on relation and on image.
• Figs 2.4 (p. 28), 2.5 and 2.6 (p. 29), 2.7 (p. 30).