Reflexive, symmetric and transitive as three demands that can fail independently

What to assume they know

• A relation as a subset of a product set, and the two extreme cases — a relation as any subset whatever of A × A, and the empty and universal relations
• Reading a relation from a rule as a list of ordered pairs on a finite set
• Congruence of triangles, and perpendicular and parallel lines in a plane
• Divisibility, parity, and the absolute value of a difference
• Comparing real numbers with the at-most sign, and squaring and cubing them
• What it takes to disprove a universally quantified claim: one counterexample

What they should be able to do

• State each of the three conditions of Definition 3 and identify how many elements of A each one talks about at a time
• Test a relation given by a rule against each condition separately, and give the witness pair or triple whenever one fails
• Explain why a single failing instance settles a condition, while confirming instances never do
• Recognise a relation with no chains at all and explain why it is transitive without any checking
• Refute the argument that symmetry together with transitivity forces reflexivity, naming the step that fails and the relation that breaks it
• Construct, on a three-element set, a relation with any prescribed pattern of pass and fail across the three conditions
• Classify a printed relation against all three conditions and choose the correct option in a multiple-choice item
• Count the relations on a small set that meet a prescribed pattern

Where it usually goes wrong

• "Symmetric plus transitive gives reflexive." The argument runs: take any a, find b with a related to b, swap to get b related to a, chain to get a related to a. The step that fails is the second word of the sentence — find. Nothing guarantees any such b exists. On {1, 2, 3} the relation {(1, 1)} passes the second and third demands and fails the first, because 2 and 3 are related to nothing at all and so are never dragged onto the diagonal.
• "A relation with almost all the diagonal pairs is nearly reflexive." Reflexivity is a demand on every element without exception. In Exercise 1.1 Q6 the relation on three elements misses all three diagonal pairs; but a relation missing only (3, 3) fails just as completely. There is no partial credit inside the definition.
• "If I have checked several pairs and they all worked, the relation is symmetric." Confirming instances never settle a universal claim; one failing instance settles it. This is why every chapter verdict of fails is delivered with a named pair, and every verdict of passes is delivered with an argument covering all cases.
• "A relation with no pairs to chain must fail transitivity." It passes. Exercise 1.1 Q1(ii) is the case: the first entries are 1, 2, 3 and the second entries are 6, 7, 8, so no pair can be continued and the demand is never put to the test. Students find this the hardest single point in §1.2 and it is worth a whole section.
• "Transitive means a is related to c whenever c is somewhere downstream." The demand is stated for exactly two steps. Longer chains follow from repeating it, but the definition itself is about a chain of length two and nothing else.
• "Perpendicular and parallel behave the same way." Perpendicularity is symmetric and fails the other two. Parallelism, in Exercise 1.1 Q14, passes all three. Fig 1.1 is precisely the picture of the difference: chain two perpendiculars and you land on a parallel.
• "These properties are about the rule, so I can read them off the wording." Exercise 1.1 Q1(iv) admits a pair when a difference is an integer — a rule that sounds restrictive and in fact excludes nothing, because the relation is on Z. The verdict depends on the set as much as on the rule.

Questions to check understanding

• Determine whether a relation given by a rule is reflexive, symmetric and transitive, justifying each verdict separately — the form of Exercise 1.1 Q1, where the marks sit in the justification
• Show that a stated relation has exactly two of the three properties, naming the witness for the failure
• Supply a relation having a prescribed pattern across the three properties — the form of Exercise 1.1 Q10
• Choose the correct classification of a listed relation from four options
• Count the relations on a three-element set satisfying stated constraints
• Explain why a particular relation is transitive when no chain exists to test

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.

• Definition 3 (§1.2, Part I p. 2). Three lettered conditions. The first is written with a quantifier over a single element of A; the second and third are written with quantifiers over two and three elements respectively, and the book names those elements with subscripts. The subscript count is itself the point of sections 2 to 4 and should be shown.
• Example 2 (Part I p. 3). T is the set of triangles in a plane; the relation admits a pair when the first triangle is congruent to the second. All three demands pass, and the chapter's reasons are one line each: a triangle is congruent to itself; congruence read backwards is still congruence; congruence chains.
• Example 3 and Fig 1.1 (Part I p. 3). L is the set of lines in a plane; the relation admits a pair when the first line is perpendicular to the second. Symmetry passes. Reflexivity fails because no line meets itself at a right angle. Transitivity fails, and the figure shows why: two horizontal lines labelled with subscripts three and one, joined by a vertical segment labelled with subscript two. The vertical line is perpendicular to both horizontals, and the two horizontals are parallel to each other, not perpendicular.
• Example 4 (Part I p. 3). On the set {1, 2, 3}, the relation {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)}. Verified: all three diagonal pairs are present, so reflexivity passes. (1, 2) is present and (2, 1) is not, so symmetry fails. (1, 2) and (2, 3) are both present and (1, 3) is not, so transitivity fails. This is the cleanest single separation in the chapter — the first demand passing while the other two fail, on five pairs.
• Exercise 1.1 Q1 (Part I p. 5), five relations to classify: (i) on {1, 2, 3, ..., 13, 14}, the pair (x, y) is admitted when 3x – y is zero; (ii) in N, admitted when y = x + 5 and x is below 4; (iii) on {1, 2, 3, 4, 5, 6}, admitted when x divides y; (iv) in Z, admitted when x – y lies in Z; (v) among the human beings of a town at one moment, five separate relations — (a) same workplace, (b) same locality, (c) x taller than y by exactly seven centimetres, (d) x is the wife of y, (e) x is the father of y. Verified: (i) the pairs are (1, 3), (2, 6), (3, 9) and (4, 12), since x = 5 would need y = 15, outside the set; it fails all three demands — (1, 1) is absent, (3, 1) is absent while (1, 3) is present, and (1, 3) with (3, 9) produces no (1, 9). (ii) the pairs are (1, 6), (2, 7) and (3, 8); it fails the first two and passes the third, because no second entry is ever also a first entry, so there is no chain to check. (iii) divisibility passes the first and third and fails the second, since 2 is divisible by 1 but not the reverse. (iv) every difference of integers is an integer, so this is the universal relation on Z and passes all three. (v)(a) and (v)(b) pass all three; (v)(c) fails all three — nobody exceeds their own height, taller reverses to shorter, and two seven-centimetre steps make fourteen; (v)(d) and (v)(e) fail all three.
• Exercise 1.1 Q2, Q4, Q5 (Part I p. 5) — three relations on R that differ only in the right-hand side. Q2 admits (a, b) when a is at most b squared; Q4 when a is at most b; Q5 when a is at most b cubed. Verified: Q4 passes reflexivity and transitivity and fails symmetry (1 is at most 2, 2 is not at most 1). Q2 fails all three — reflexivity dies at a = one half, since one half exceeds one quarter; symmetry dies on (1, 2) against (2, 1); transitivity dies on the triple 10, 4, 2, because 10 is at most 16 and 4 is at most 4, yet 10 exceeds 4. Q5 fails all three by the same shape — reflexivity at a = one half again, symmetry on (1, 2) against (2, 1), and transitivity on the triple 10, 3, 2, because 10 is at most 27 and 3 is at most 8, yet 10 exceeds 8. Putting Q2, Q4 and Q5 next to each other is the single most efficient thing the explanation can do: one symbol changes and the verdict changes twice.
• Exercise 1.1 Q3 (Part I p. 5). On {1, 2, 3, 4, 5, 6}, admitted when b = a + 1. Verified: the pairs are (1, 2), (2, 3), (3, 4), (4, 5), (5, 6); all three demands fail.
• Exercise 1.1 Q6 (Part I p. 6). On {1, 2, 3}, the two-pair relation {(1, 2), (2, 1)}. Verified: symmetry passes. Reflexivity fails on every element, 3 included. Transitivity fails because (1, 2) with (2, 1) demands (1, 1), which is absent. This relation is the counterweight to the false proof of section 8 — swapping alone buys nothing.
• Exercise 1.1 Q10 (Part I p. 6). Five patterns for which the student must supply a relation. Verified, one workable answer each, all on {1, 2, 3}: second only — {(1, 2), (2, 1)}; third only — {(1, 2)}; first and second but not third — {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}; first and third but not second — {(1, 1), (2, 2), (3, 3), (1, 2)}; second and third but not first — {(1, 1)}.
• The eighth pattern, and the smallest set that carries them all (not in the book). All three passing is the universal relation; all three failing is {(1, 2), (2, 3)}. Together with the six specimens above that is all eight patterns on a three-element set. Verified: on a two-element set the spread is not available — the only relations passing both the first and second demands there are {(1, 1), (2, 2)} and the universal one, and both are transitive, so the first-and-second-but-not-third pattern cannot occur. Three elements is the smallest arena in which independence is fully visible, and that is worth saying.
• Exercise 1.1 Q15 (Part I p. 7). On {1, 2, 3, 4}, the relation {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}, with four options. Verified: all four diagonal pairs are present, so reflexivity passes. (1, 2) is present without (2, 1), so symmetry fails. Every chain closes — the one to check is (1, 3) with (3, 2), which needs (1, 2), and it is there. So the relation is reflexive and transitive and not symmetric.
• Miscellaneous Example 23 (Part I p. 14). On {1, 2, 3}, count the relations containing (1, 2) and (2, 3) that pass the first and third demands and fail the second. The chapter works this through and reaches three. The argument to reproduce is the one that matters: adding certain pairs forces others in for transitivity, and the forcing eventually makes the relation symmetric, which disqualifies it.
• Miscellaneous Exercise Q6 (Part I p. 16). On {1, 2, 3}, count the relations containing (1, 2) and (1, 3) that pass the first and second demands and fail the third. Four options are offered. Verified: reflexivity forces the three diagonal pairs; symmetry forces (2, 1) and (3, 1); that seven-pair relation already fails transitivity, since (2, 1) with (1, 3) needs (2, 3), which is absent — so it qualifies. The only pairs left are (2, 3) and (3, 2), and symmetry means they must be added together; adding both gives all nine pairs, which is transitive and therefore disqualified. Exactly one relation qualifies.

Figures to have open

• The grid of A × A with a marked diagonal and a reflection line. This is an added device and carries sections 2, 3 and 4; it should be the same grid used in the previous topic so the two videos read continuously. Standard schematic.
• A redraw of Fig 1.1 (Part I p. 3): two horizontal parallel lines with a vertical segment joining them, the three lines labelled with subscripts one, two and three as printed, plus right-angle marks the textbook figure does not draw. From the textbook's figure, redrawn.
• A right-angled triangle and a scaled copy for section 5. Standard schematic.
• An eight-cell table of pass-and-fail patterns for section 11. Not in the book; the chapter never assembles one.

Where this sits in the book

• NCERT Class 12 Mathematics, Part I, Chapter 1 "Relations and Functions", §1.2 Types of Relations, Definition 3, p. 2, and Definition 4, p. 3
• Examples 2, 3 and 4 with Fig 1.1, p. 3
• Exercise 1.1, questions 1 to 6, 10 and 15, pp. 5 to 7
• Miscellaneous Example 23, p. 14, and Miscellaneous Exercise question 6, p. 16