Why order and repetition cannot make two sets different

What to assume they know

• Listing the members versus stating the property they share — roster form, set-builder form, and the distinctness rule
• A set with nothing in it, and sets you cannot finish listing — the empty set, and finite versus infinite
• Prime numbers and prime factorisation of a two-digit number
• Solving a quadratic by factorising, including one with roots of opposite sign
• Reading a condition that combines two inequalities

What they should be able to do

• State the condition under which two sets are equal, as a check running in both directions
• Show that reordering a roster leaves the set unchanged, and give the reason in terms of the membership question
• Show that deleting repeats leaves the set unchanged, for the same reason
• Decide equality for two sets given by different kinds of description — one a list, one a condition
• Produce a counterexample when two sets are unequal, by naming a single member of one that fails in the other
• Recognise when a set is empty during an equality check, and use that to settle several comparisons at once
• Write the not-equal statement correctly and say what it takes to justify it

Where it usually goes wrong

• "Two sets are equal when they are described the same way." The primes below 6 and the prime factors of 30 share no wording at all and are the same set.
• "Two sets are unequal when they are described differently." Same example, read the other way.
• "Checking one direction is enough." One direction gives containment, not equality; the chapter's own subset section makes this explicit later. A student who checks only that every member of the first is in the second has proved something weaker.
• "{–5, 5} and {5} are nearly the same." They differ by a member, so they are unequal, full stop. Example 7 is built on this.
• "A longer word must give a bigger letter set." CATARACT has eight letters and four distinct ones; TRACT has five and the same four.
• "To disprove equality I must list both sets fully." One member on one side that fails on the other is a complete proof. Example 8(ii) is settled by 0.
• "An empty set is equal to nothing at all." It is equal to every other empty set and unequal to every non-empty one — which is exactly why B settles four comparisons in Example 7.
• "F = {0, a} and H = {0, 1} might be equal." Only if a were 1, and no part of the question says so. Treat an unbound letter as unknown, not as convenient.

Questions to check understanding

• Decide whether two given sets are equal, and justify — the standard two-mark form, with the justification carrying the marks
• Select the equal pairs from a printed group of six or eight sets
• Show two sets equal where one is a list and the other a condition
• Disprove an equality by exhibiting one member, and say why one is enough
• Reduce two words to their letter sets and compare

Examples worth working on the board

Values marked verified are an added derivation from the chapter's own data; the chapter prints no answers to its exercises.

• The scrambled roster (§1.5, p. 7). {1, 2, 3, 4} is set beside {3, 1, 4, 2} and the two are equal. Run the check live in section 3: pick each of the four members in turn and find it on the other side, then do the reverse.
• Two unrelated descriptions, one set (§1.5, p. 7). The primes below 6 are compared with the prime factors of 30. Verified: the primes below 6 are 2, 3 and 5; and 30 = 2 × 3 × 5, so its prime factors are the same three. Neither description mentions the other, and the sets are equal — this is section 5.
• The repeat (§1.5, pp. 7–8, boxed Note). {1, 2, 3} is set beside a list in which 2 and 3 each appear twice, and the two are equal. The chapter's stated reason is that each member of one is a member of the other, both ways round.
• Example 7 (p. 8) — five sets, and the task is to find every equal pair. A = {0}; B is the set of things bigger than 15 and smaller than 5; C is defined by x – 5 = 0; D by x² = 25; E is the positive integer root of x² – 2x – 15 = 0. Verified: nothing is both above 15 and below 5, so B is empty; C = {5}; x² = 25 gives D = {–5, 5}; and x² – 2x – 15 factorises as (x – 5)(x + 3), whose roots are 5 and –3, of which only 5 is a positive integer, so E = {5}. Hence the only equal pair is C and E. Note how much work the empty set does here: once B is known to be empty it is unequal to all four others at a stroke, and A is settled by the single member 0.
• Example 8 (p. 8) — two pairs to judge. (i) the letters of ALLOY against the letters of LOYAL. (ii) A is the integers with n² ≤ 4, and B is the real numbers satisfying x² – 3x + 2 = 0. Verified: both words use exactly A, L, O and Y, so (i) is a pair of equal sets even though the letters are ordered differently and L is doubled in each; for (ii), A = {–2, –1, 0, 1, 2} and the quadratic factorises as (x – 1)(x – 2) so B = {1, 2}, and 0 sits inside A while B excludes it — one member settles it.
• Exercise 1.2 Q4 (p. 9) — four pairs to be judged equal or not. (i) a, b, c, d against d, c, b, a. (ii) {4, 8, 12, 16} against {8, 4, 16, 18}. (iii) {2, 4, 6, 8, 10} against the positive even integers up to 10. (iv) the multiples of 10 against a list beginning 10, 15, 20, 25, 30 and continuing. Verified: (i) equal, order only; (ii) unequal, since 12 is in the first and 18 in the second; (iii) equal; (iv) unequal — 15 is in the listed set and is not a multiple of 10.
• Exercise 1.2 Q5 (p. 9) — two more pairs. (i) {2, 3} against the solutions of x² + 5x + 6 = 0. (ii) the letters of FOLLOW against the letters of WOLF. Verified: the quadratic factorises as (x + 2)(x + 3), giving –2 and –3, so the first pair is unequal and the sign is the whole trap; the second pair is equal, both reducing to F, O, L, W.
• Exercise 1.2 Q6 (p. 9) — eight sets to be sorted into equal families: A = {2, 4, 8, 12}; then B = {1, 2, 3, 4}; C = {4, 8, 12, 14}; next D = {3, 1, 4, 2}; E = {–1, 1}; then F = {0, a}; G = {1, –1}; and last H = {0, 1}. Verified: B and D are equal and E and G are equal; A and C differ at 2 and 14; F and H differ unless a is taken to be 1, which nothing states. Two families, and four sets left alone.
• Miscellaneous Example 23 (p. 20, running to p. 21). The letters needed for CATARACT are compared with those needed for TRACT. Verified: both reduce to C, A, T and R, so the sets are equal — an eight-letter word and a five-letter word naming one four-member set.

Figures to have open

• A two-arrow equality diagram: a left set and a right set with one arrow labelled every member of this is in that and its mirror. Standard schematic; the chapter states the condition in prose.
• A collapsing-roster movement frame set for section 4, showing repeats being absorbed rather than deleted. Standard schematic.
• A word-to-letter-set reducer used three times over — SCHOOL from the earlier topic, then ALLOY and LOYAL, then CATARACT and TRACT. Standard schematic.
• No textbook figure is needed; the chapter's numbered figures begin at Fig 1.1 on p. 11.

Where this sits in the book

• NCERT Class 11 Mathematics, Chapter 1 "Sets", §1.5 Equal Sets, pp. 7–8, including Definition 3 and the boxed Note that runs across the page break
• Worked Examples 7 and 8, p. 8
• Exercise 1.2, questions 4 to 6, p. 9
• Miscellaneous Example 23, pp. 20–21, which is an equality question and belongs with this topic even though it sits at the end of the chapter