Union and intersection, and what it means for two sets to miss each other entirely

What to assume they know

• Turning a claim about sets into a picture you can read off — Venn diagrams and how a shading is read
• Why nothing can be complemented until the surrounding set is fixed — the universal set, assumed fixed from §1.9 onwards
• Why order and repetition cannot make two sets different — equality of sets checked in both directions
• Containment, proper containment, and why the empty set is inside everything — containment
• Prime numbers, even and odd natural numbers

What they should be able to do

• Form the union and the intersection of two given sets, in roster form
• State each operation in set-builder form and name the connective it encodes
• Explain why an element common to both sets is listed once in the union
• Predict the union and the intersection when one set is contained in the other, and prove the prediction
• State the listed properties of each operation and give the reason for each in terms of its connective
• Decide whether two given sets are disjoint, and justify the verdict
• Apply the distributive law and verify it on given sets
• Interpret a union or intersection in a real context and say what the resulting set means

Where it usually goes wrong

• "The union of a four-member set and a four-member set has eight members." Example 12 gives six. Membership is a verdict, not a tally; a shared object is one object.
• "Union means adding the sets." Addition on numbers has no idempotent law; union does, precisely because or repeated is still or.
• "Intersection is the smaller set." It is the set of shared members, which may be smaller than both, or equal to one of them, or empty. Example 17 gives the case where it equals one of them.
• "Disjoint means unequal." It means no shared member. Two unequal sets can overlap heavily.
• "An empty intersection means one of the sets is empty." {2, 4, 6, 8} and {1, 3, 5, 7} are both non-empty and share nothing.
• "The identity element is the same for both operations." The empty set leaves a union alone; the universe leaves an intersection alone. The two swap, and the reason is that or false and and true are the harmless cases.
• "The distributive law works the way it does in arithmetic, so only one version is worth learning." The chapter prints the version where intersection spreads across union and checks it in five panels.
• "Union and intersection can be read off two sets without fixing a universe." For the operations themselves, yes; but the chapter states at the head of §1.9 that every set from there on is taken inside a universe, and the property involving the universe depends on it.

Questions to check understanding

• Form the union and intersection of two sets given in roster or set-builder form
• Given a containment, state the union and intersection without computing
• Decide whether given pairs are disjoint, with justification
• Verify the distributive law on three stated sets
• Interpret a union or intersection in a stated real context
• Evaluate a compound expression such as an intersection of two unions

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 chapter's framing (§1.9, pp. 13–14). Addition takes 5 and 13 to 18; multiplication takes the same pair to 65. Two numbers in, one number out — and the section then asks for operations that take two sets in and give one set out. Use the two arithmetic facts as printed; they are the hook.
• Example 12 (p. 14). A = {2, 4, 6, 8} and B = {6, 8, 10, 12}. Verified: the union holds 2, 4, 6, 8, 10, 12 — six members, not eight, because 6 and 8 are shared. Verified: the intersection is {6, 8} (Example 15, p. 15). One pair of sets carries both operations, which is how.
• Example 13 (p. 14). A is the five vowels and B holds a, i and u. Verified: the union is A itself, because every member of B was already in A. The chapter draws the general moral: when one set sits inside another, the union is the larger.
• Example 14 (p. 14). X is a hockey team of three named Class XI students and Y a football team of three, sharing one name. Verified: the union has five members and means the students on at least one of the two teams; the intersection has one member and means the student on both (Example 16, p. 15). This pair is the best interpretation exercise in the section.
• Example 17 (p. 15). A is 1 to 10 and B = {2, 3, 5, 7}. Verified: B holds exactly the primes up to 10, so B sits inside A and the intersection is B — the mirror of Example 13.
• Union's properties (§1.9.1, pp. 14–15), printed as five: order does not matter; grouping does not matter; the empty set is the identity; a set unioned with itself is unchanged; and the universe absorbs everything.
• Intersection's properties (§1.9.2, pp. 15–16), printed as five: order does not matter; grouping does not matter; the empty set absorbs and the universe is the identity; a set met with itself is unchanged; and intersection distributes across union.
• Disjoint sets (§1.9.2, p. 15). A = {2, 4, 6, 8} and B = {1, 3, 5, 7} share nothing. Verified: every member of the first is even and every member of the second odd, so no object can satisfy both conditions — the reason, not just the observation.
• Fig 1.4 (p. 14) and Fig 1.5 (p. 15). Read off the printed pages: Fig 1.4 shows two overlapping circles inside a rectangle labelled U, both circles tinted, with the union named below them; Fig 1.5 shows the same arrangement with only the overlapping lens tinted and the intersection named beneath it, an arrow pointing up into the lens. Fig 1.6 (p. 15) shows two untouching circles, labelled A and B and neither tinted, drawn inside the rectangle for the universe.
• Exercise 1.4 Q1 and Q5 (p. 17) — five pairs, each to be unioned and then intersected. Verified: {1, 3, 5} with {1, 2, 3} gives union {1, 2, 3, 5} and intersection {1, 3}; the vowels with {a, b, c} give a union holding all five vowels together with b and c, and intersection {a}; the multiples of 3 with the naturals below 6 give a union holding 1, 2, 3, 4, 5 and then 6, 9, 12 and onwards, and intersection {3}; the naturals above 1 and at most 6 with the naturals strictly between 6 and 10 give union {2, 3, 4, 5, 6, 7, 8, 9} and empty intersection; and {1, 2, 3} with the empty set gives union {1, 2, 3} and empty intersection.
• Exercise 1.4 Q2 and Q3 (p. 17). With A = {a, b} and B = {a, b, c}, A sits inside B and the union is B; question 3 asks for the general statement, which is the moral of Example 13.
• Exercise 1.4 Q4 (p. 17). A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8}, D = {7, 8, 9, 10}, with seven unions to form, three of them of three sets at once. Verified: the three-set unions are where associativity earns its keep — A with B with C is 1 to 8.
• Exercise 1.4 Q6 (p. 17). A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}, with ten intersections. Verified: A with B is {7, 9, 11}; B with C is {11, 13}; A with C is {11}; A with D and B with D are both empty; A with C with D is empty; A with the union of B and C is {7, 9, 11}; A with the union of B and D is {7, 9, 11}; the intersection of A-with-B and B-with-C's union is {7, 9, 11}; and the union of A and D met with the union of B and C is {7, 9, 11, 15}. The last of these is the only one where D contributes, and it is worth doing slowly.
• Exercise 1.4 Q7 (p. 18). A is the naturals, B the even naturals, C the odd naturals, D the primes. Verified: A meets each of the others in the smaller set; B and C are disjoint; B meets D in {2} alone; C meets D in every prime except 2. That last pair is the best item in the exercise.
• Exercise 1.4 Q8 and Q12 (p. 18). Pairs to be tested for disjointness, and four true-or-false statements about disjointness. Verified: {1, 2, 3, 4} and the naturals from 4 to 6 share 4, so they are not disjoint; the vowels and {c, d, e, f} share e; the even and odd integers are disjoint; among the true-or-false items, {2, 3, 4, 5} and {3, 6} share 3, the vowels and {a, b, c, d} share a, and the two remaining pairs are genuinely disjoint.

Figures to have open

• Redraws of Fig 1.4 (p. 14) and Fig 1.5 (p. 15): two overlapping circles in a rectangle, tinted whole in the first and only across the lens in the second. These are the chapter's own figures; redraw as clean schematics.
• A redraw of Fig 1.6 (p. 15) for section 8: the rectangle for the universe with two untouching labelled circles inside it. Keep the rectangle — the empty overlap is only readable against the surrounding region.
• A redraw of Figs 1.7 (i) to (v) (p. 16) for section 10, keeping the chapter's panel order and labelling.
• A two-column property board pairing each printed law with its connective. An added standard schematic; the chapter lists the laws and does not motivate them.

Where this sits in the book

• NCERT Class 11 Mathematics, Chapter 1 "Sets", §1.9 Operations on Sets, pp. 13–14, §1.9.1 Union of sets, pp. 14–15 with Definition 5, and §1.9.2 Intersection of sets, pp. 15–16 with Definition 6
• Worked Examples 12, 13, 14, 15, 16 and 17, pp. 14–15
• Fig 1.4, p. 14; Fig 1.5 and Fig 1.6, p. 15; Figs 1.7 (i) to (v), p. 16
• Exercise 1.4, questions 1 to 8 and question 12, pp. 17–18
• Miscellaneous Exercise questions 3, 4, 6, 7 and 8, pp. 21–22, which are the proof-flavoured versions of this topic