A relation as a subset of a product set, and the two extreme cases

What to assume they know

• Ordered pairs, and the product of two sets written A × B (Class XI)
• Reading and writing a set two ways — by listing its members, and by a rule
• Subset, and that the empty set is a subset of every set
• The absolute value of a real number, and that it is never negative
• Divisibility and simple integer arithmetic
• The number systems N, Z and R, and their standing symbols

What they should be able to do

• State the chapter's definition of a relation carried by one set, and apply it as a membership test on a stated pair
• Convert a relation given by a rule into its list of ordered pairs on a small finite set, and back
• Use both notations for the same fact — the ordered pair sitting in R, and the infix form with R between the two symbols
• Show that a stated rule selects no pair at all, and identify the resulting relation as the empty one
• Show that a stated rule selects every pair, and identify the resulting relation as the universal one
• Explain why both extremes must be admitted as relations rather than dismissed, and why the chapter groups them under one name
• Distinguish a relation from A to B from a relation in a single set A, and say which one §1.2 works with
• Count the ordered pairs available on a small finite set, and say how many relations that permits

Where it usually goes wrong

• "A relation has to mean something." The five opening examples all mean something, and the definition that replaces them means nothing at all. The book says explicitly that it does not require a recognisable link. Any set of pairs qualifies, including one chosen at random.
• "The empty relation is a degenerate case, not really a relation." It is named, defined and numbered — Definition 1. It qualifies because the empty set sits inside A × A as a subset, and every subset is a relation. Excluding it would make the definition of a relation conditional, which is exactly what the chapter is trying to avoid.
• "A rule that selects nothing must be a badly written rule." The rule a – b = 10 is perfectly well written; it is the set that makes it select nothing. Move to A = {1, 2, ..., 20} and the same rule selects ten pairs. The relation depends on the rule and the set together, never on the rule alone.
• "Universal relation means the two sets are the same size, or that A = B." Universal is about one set A and means the relation is the whole of A × A. Size and equality of two different sets are not in the definition at all.
• "An absolute value could come out negative if a is smaller than b." It cannot; that is what the bars do. This is why the second rule on Part I p. 2 admits every pair without a single case check, and it is worth pausing on, because students often verify it on two or three pairs and never see that it needed no verification.
• "a R b and (a, b) belonging to R are two different facts." They are one fact in two notations, and the chapter says so in its Remark. Students who learn only one form stall when an exercise uses the other.
• "Relations are between two different sets, so a relation in one set is unusual." §1.1 sets up two different sets and §1.2 immediately narrows to one. Everything the chapter goes on to define — reflexive, symmetric, transitive, equivalence — is defined only for a relation in a single set, because those words need to compare an element with itself.

Questions to check understanding

• Given a set and a rule, write the relation in roster form and count its pairs
• Show that a stated relation is the empty relation, or that it is the universal relation, and justify from the rule rather than by listing
• Decide which of four given ordered pairs belongs to a relation defined by two simultaneous conditions — the form of Exercise 1.1 Q16
• Rewrite a relation given in infix form as a set of ordered pairs, and the reverse
• Construct a rule that gives the empty relation on one stated set and a non-empty relation on another
• Explain, without listing, why a stated rule admits every pair of a set

Examples worth working on the board

Values marked verified are worked out here on the chapter's own data; no answer key was consulted, and the chapter prints no answers to its exercises.

• The five opening relations (§1.1, Part I p. 1). A is the set of Class XII students of a school, B the set of Class XI students of the same school. The five links offered are: b has a for a brother; b has a for a sister; a's age exceeds b's; a's total in the final examination falls short of b's; the two of them share a locality. All five are then discarded as motivation only — the definition that follows keeps none of the link, only the subset.
• The chapter's own disclaimer (§1.1, Part I p. 1). Immediately after the definition the book says outright that it does not care whether any recognisable connection exists between the two entries of a pair. Section 3 of the explanation is built on this one sentence's content.
• The two extreme rules on four elements (§1.2, Part I p. 2). A = {1, 2, 3, 4}. First rule: the pair (a, b) is admitted when a – b equals ten. Second rule: the pair is admitted when the absolute value of a – b is at least zero. Verified: on this A the largest available difference is 4 – 1 = 3 and the smallest is 1 – 4 = –3, so no pair meets the first rule and the relation is empty. An absolute value is never negative, so every pair meets the second rule unconditionally and the relation is the whole of A × A.
• The size of the ground being swept (not in the book). With four elements, A × A holds 4 × 4 = 16 ordered pairs, so there are 2 to the power 16, that is 65536, distinct relations in A. The chapter states neither figure; use them only to show that the two extremes sit at the ends of something very large.
• Definition 1 and Definition 2 (§1.2, Part I p. 2). The first names the relation in which nothing is related to anything and records it as the empty set contained in A × A. The second names the relation in which everything is related to everything and records it as A × A itself. Both are then called trivial relations.
• Example 1 (§1.2, Part I p. 2). A collects every pupil at a school that admits boys only. The first relation admits (a, b) when b has a for a sister; the second admits (a, b) when the heights of a and b differ by less than three metres. Verified: the first is empty because the school has no girls, so no pair can ever qualify; the second is universal because no two human heights differ by three metres. Note that both verdicts come from facts about the world, not from algebra — which is the point of putting this example next to the four-element one.
• Reading a rule as a list (Remark, §1.2, Part I p. 2). On {1, 2, 3, 4} with the rule b = a + 1. Verified: the pairs are (1, 2), (2, 3) and (3, 4); a = 4 is excluded because 5 is not in the set. Three pairs out of the sixteen available.
• The same rule on a longer set (Exercise 1.1 Q3, Part I p. 5). On {1, 2, 3, 4, 5, 6} with the rule b = a + 1. Verified: five pairs — (1, 2), (2, 3), (3, 4), (4, 5), (5, 6).
• A membership question settled pair by pair (Exercise 1.1 Q16, Part I p. 7). R is a relation in N admitting (a, b) exactly when a = b – 2 and b > 6. The four candidates offered are (2, 4), (3, 8), (6, 8) and (8, 7). Verified: (2, 4) fails because 4 is not greater than 6; (3, 8) fails because 8 – 2 is 6, not 3; (8, 7) fails because 7 – 2 is 5, not 8; (6, 8) passes both conditions. So the pair in R is (6, 8). Note that two separate conditions have to be checked for each candidate, and each of the three failures fails a different one.
• A rule whose relation turns out to be universal (Exercise 1.1 Q1 item iv, Part I p. 5). In Z, the pair (x, y) is admitted when x – y is an integer. Verified: the difference of two integers is always an integer, so nothing is excluded and this relation is the whole of Z × Z — the universal relation of Definition 2, arriving in an exercise where it is not announced as one.
• The chapter's frontispiece (Part I p. 1). The opening page carries an epigraph attributed to G. H. Hardy about mathematical beauty, a portrait captioned Lejeune Dirichlet with the dates 1805–1859, and a QR code marked with the Part I catalogue number.

Figures to have open

• A grid of the sixteen cells of A × A for A = {1, 2, 3, 4}, shadable cell by cell. This is an added device and carries sections 3, 5, 6 and 7. It must be the same grid throughout so that the empty and universal relations are visibly the two ends of one range. Standard schematic.
• Two class lists with arrows drawn between them, for section 1. Standard schematic; the chapter draws nothing here.
• The opening page's Dirichlet portrait and Hardy epigraph (Part I p. 1) are the textbook's own. Use a plain caption card naming the portrait and its dates rather than reproducing the printed page. The QR code on the same page should be ignored.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 1 "Relations and Functions", §1.1 Introduction, p. 1
• §1.2 Types of Relations, opening paragraph, Definitions 1 and 2, Example 1 and the Remark that follows it, p. 2
• Exercise 1.1, questions 1(iv), 3 and 16, pp. 5 and 7