Containment, proper containment, and why the empty set is inside everything

What to assume they know

• Why order and repetition cannot make two sets different — equality of sets as a check running both ways
• A set with nothing in it, and sets you cannot finish listing — the empty set
• The belongs-to symbol and its negation
• Divisors and prime divisors of a two-digit number
• Reading an implication arrow, and the idea that an implication with no case to test is not violated

What they should be able to do

• State the containment condition as an implication about an arbitrary member
• Decide containment between two given sets and justify the verdict either by the implication or by one failing member
• Explain why containment holding in both directions is the same statement as equality, and use the two-way symbol correctly
• Explain why every set contains itself, and why the empty set is contained in every set
• Distinguish a subset from a proper subset, and name the superset in a given pair
• Use the belongs-to and contained-in symbols correctly when the members of a set are themselves sets
• Show by example that belonging followed by containment does not give containment
• List every subset of a set with one, two or three members, and say how many there should be

Where it usually goes wrong

• "Subset means smaller." Every set is a subset of itself, and the chapter says so before it introduces the word proper. Smaller is what proper adds.
• "The empty set is a subset because it is too small to cause trouble." The actual reason is that the implication has no case to test: there is no member of the empty set that could fail to be in the other set. Note that the chapter presents this as something we agree to say rather than as a proved statement.
• "Belonging and containment are two spellings of one idea." They take different things on the left: an object, and a set. Example 11 shows they do not chain, and Exercise 1.3 Q3 is eleven repetitions of the same warning.
• "If A is inside C and B is inside C then one of A, B is inside the other." Example 9 refutes it: {1, 3} and {1, 5, 9} both sit inside {1, 3, 5, 7, 9} and neither sits inside the other.
• "{3, 4} is inside a set that has {3, 4} as a member." Containment would need 3 and 4 themselves to be members. They are not.
• "The empty set is a member of every set." It is a subset of each set there is. A set has the empty set as a member only if it was put there.
• "A one-member set is the same as its member." The set is a container; the fifth part of Exercise 1.3 Q2 is built on exactly this confusion.
• "Listing subsets is guesswork." Each member is independently in or out, so the count is fixed in advance — 8 for a three-member set — and a list that gives seven or nine is wrong before it is read.

Questions to check understanding

• Fill in the containment symbol or its negation between two described sets
• True or false with justification, on mixed statements using belonging and containment
• Identify which of a list of statements about a nested set are incorrect, and say why — the exact form of Exercise 1.3 Q3
• Write down every subset of a given small set
• Arrange several given sets by containment
• Prove or disprove: if A is a member of B and B sits inside C, must A sit inside C

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 opening pair (§1.6, p. 9). X is every student in a school and Y is every student in one class of it. Y sits inside X. Use this before any symbol appears.
• Four containment checks (§1.6, p. 10). (i) the rationals inside the reals; (ii) the prime divisors of 56 inside all its divisors; (iii) {1, 3, 5} against the odd naturals below 6; (iv) the vowels against {a, b, c, d}. Verified: 56 = 2³ × 7, so its divisors are 1, 2, 4, 7, 8, 14, 28, 56 and its prime divisors are 2 and 7 — eight against two. In (iii) both containments hold, so the sets are equal. In (iv) neither holds: e is outside the second set and b is outside the first, so one failing member each way is the entire proof.
• Proper containment (§1.6, p. 10). {1, 2, 3} sits properly inside {1, 2, 3, 4}, and the chapter names the second the superset of the first.
• Singleton (§1.6, p. 10). A set holding only a is a singleton set.
• Example 9 (p. 10). Four containment questions among the empty set, A = {1, 3}, B = {1, 5, 9} and C = {1, 3, 5, 7, 9}. Verified: the empty set is inside B; A is not inside B, and 3 is the member that proves it; A is inside C; and B is inside C. Notice that A and B are both inside C but neither is inside the other — containment does not order everything.
• Example 10 (p. 10). The vowels and {a, b, c, d} again, asked as two questions, each answered no. The chapter asks the student to supply the reason.
• Example 11 (p. 10) — the pivot of the topic. A = {1}; B has two members, the set {1} and the number 2; C has three, the set {1}, the number 2 and the number 3. Then A is a member of B, and B sits inside C, yet A is not inside C. Verified: the reason is that 1 belongs to A, and 1 is not one of C's three members — C holds the set {1}, not the number 1. Draw the three levels.
• Exercise 1.3 Q1 (p. 12) — seven pairs to be joined by the containment symbol or its negation: {2, 3, 4} and {1, 2, 3, 4, 5}; {a, b, c} and {b, c, d}; the Class XI students of your school and all its students; the circles in a plane and the unit-radius circles in that plane; the triangles in a plane and the rectangles in it; the equilateral triangles in a plane and the triangles in it; the even naturals and the integers. Verified: containment holds in the first, third, sixth and seventh; it fails in the second (a is outside), the fourth (a circle of radius 2 is outside) and the fifth (a triangle is not a rectangle). Item four is a deliberate reversal — the wider family is written second.
• Exercise 1.3 Q2 (p. 12) — six true-or-false statements. Verified: the first is false, since {a, b} does sit inside {b, c, a}; the second and fourth are true; the third is false because 2 is missing from {1, 3, 5}; the fifth is false because a is a member of {a, b, c} while the singleton holding a is not; the sixth is true, since the even naturals below 6 are 2 and 4 and both divide 36.
• Exercise 1.3 Q3 (p. 12) — eleven statements about the four-member set whose third member is itself the set {3, 4}. Verified: correct are — {3, 4} is a member; the set holding {3, 4} sits inside; 1 is a member; {1, 2, 5} sits inside; and the empty set sits inside. Incorrect are — {3, 4} sits inside (3 and 4 are not themselves members); 1 sits inside (1 is not a set); {1, 2, 5} is a member; {1, 2, 3} sits inside (3 is not a member); the empty set is a member; and the set holding the empty set sits inside.
• Exercise 1.3 Q4 (p. 12) and Miscellaneous Example 24 (p. 21). List every subset of {a}; of {a, b}; of {1, 2, 3}; and of the empty set. The worked example does the same job for {–1, 0, 1} and prints all eight. Verified: the counts are 2, 4, 8 and 1 — each member is independently in or out, so a set with n members has 2ⁿ subsets, and the empty set has exactly one, itself.
• Miscellaneous Exercise Q1 (p. 21). Four sets to be arranged by containment: the real solutions of x² – 8x + 12 = 0; {2, 4, 6}; the even naturals written with dots; and {6}. Verified: the quadratic factorises as (x – 2)(x – 6), so the first set is {2, 6}; then {6} sits inside {2, 6}, which sits inside {2, 4, 6}, which sits inside the even naturals.

Figures to have open

• Nested-boundary diagrams for sections 1, 5 and 6. The chapter draws no Venn diagram this early — its first numbered figure of that kind is Fig 1.2 on p. 13 — so these are an added standard schematics.
• A three-level box diagram for Example 11, in which the inner label is visibly a set rather than a number. This carries the whole topic and must be drawn carefully: the difference between the number 1 and the set holding 1 has to be visible at a glance.
• A two-choice tree for section 10: three members, each in or out, eight leaves. Standard schematic; the chapter lists the subsets and draws no tree.

Where this sits in the book

• NCERT Class 11 Mathematics, Chapter 1 "Sets", §1.6 Subsets, pp. 9–10, including Definition 4
• Worked Examples 9, 10 and 11, p. 10
• Exercise 1.3, questions 1 to 4, p. 12
• Miscellaneous Example 24 and Miscellaneous Exercise question 1, p. 21
• Miscellaneous Exercise question 2, p. 21, six true-or-false statements mixing belonging and containment, which is this topic's assessment in its hardest form