"Or", "and" and "not" are the three set operations under new names

What to assume they know

• Every description of a happening picks out a part of the outcome list — an event is any subset of the sample space
• The two extreme cases, and the difference between one outcome and many — the extreme events, and events sized by sample point count
• Union, intersection, difference and complement, and the Venn pictures for them, from the chapter on sets earlier in this book
• That the complement of a set is taken relative to a stated universal set
• Which of 1 to 6 are prime, and which are odd
• Writing the outcomes of two throws of a die as ordered pairs

What they should be able to do

• Write the complement of a stated event as an explicit list, and check that the two lists together exhaust the sample space
• Translate a description containing "or" into a union, stating that outcomes satisfying both descriptions are included
• Translate a description containing "and" into an intersection, and explain that a single outcome must satisfy both descriptions at once
• Translate a description of the form "the first but not the second" into a difference, and rewrite that difference using a complement and an intersection
• Carry out all four operations on one small sample space and verify the results against each other by counting
• Explain why an exclusive reading of "or" would break the correspondence between descriptions and subsets
• Identify which of the listed operations are definable from the others

Where it usually goes wrong

• "Or means one or the other, not both." The chapter states outright that the union takes outcomes lying in either set or in both. Under an exclusive reading the union of the primes and the odds on a die would lose 3 and 5, and no set operation would answer to the description at all.
• "The complement is the opposite kind of outcome." It is only everything the event omits. The three-coin complement lumps the no-tail outcome with the all-tail outcome, which are as unalike as the sample space allows.
• "A but not B removes all of B." It removes only what B shares with A. In Example 1 the second set has three members and the difference discards two of them, because the third was never in A.
• "A and B means it happens twice, or on two trials." One performance yields one outcome, and that single outcome has to satisfy both descriptions. The two-throw example still describes one performance — two throws are what the single outcome records.
• "1 is prime." It is not, and Example 1's answers change if it is treated as prime: the intersection and the union both move.
• "A difference works either way round." Taking A but not B and taking B but not A give different sets — {2} and {1} in Example 1.
• "You need all four operations to say everything." The chapter supplies the identity that builds the difference from the other two, and on p. 304 it converts a complemented union into an intersection of complements, naming the law it is using. Two suffice: complement together with either union or intersection.
• "The complement depends on which events you are discussing." It is taken against the sample space, which is fixed by the experiment, not by the events.

Questions to check understanding

• Given an experiment and two events, write the union, the intersection, the two differences and both complements as explicit lists — Exercise 14.1 Q6 sets exactly this for two thrown dice, using three events: the first die shows an even face, the first die shows an odd face, and the two faces total 5 or less. It then asks for eight combinations, one of them built from two complements and an intersection
• Exercise 14.1 Q2 ends by requiring nine combinations of six named events on a single die throw, which is the drill in its densest form
• Rewrite a difference using a complement and an intersection, and verify by listing
• Decide whether a description in words needs a union or an intersection, with distractors that use "or" where both descriptions can hold together
• Given two events, state whether one difference equals the other and justify

Examples worth working on the board

Values marked verified are worked out here against the printed page; the chapter supplies the sets and none of the counts.

• Three tossed coins, and a lone tail (§14.1.3, p. 291). The outcome list is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. The event described as exactly one tail turning up is A = {HTH, HHT, THH}, and the chapter writes its complement out as A′ = {HHH, HTT, THT, TTH, TTT}. Verified: 3 and 5, adding to the 8 outcomes, with no outcome in both. Note: the complement is not a family with anything in common — it holds the no-tail outcome, the three two-tail outcomes and the all-tail outcome together, joined only by failing A.
• The chapter's illustrative failure. Verified: HTT is not in A, so when HTT turns up the event fails and its complement occurs. The chapter uses that single outcome to motivate the whole definition.
• Two throws of a die (§14.1.3, p. 292). Outcomes are ordered pairs. A is described by a six on the first throw and B by a total of at least 11. A = {(6,1), (6,2), (6,3), (6,4), (6,5), (6,6)} and B = {(5,6), (6,5), (6,6)}, so A ∩ B = {(6,5), (6,6)}. Verified: six pairs, three pairs and two pairs; the only totals reaching 11 are 11 and 12, and exactly three pairs deliver them. Verified, not printed: the union holds 7 pairs, since 6 + 3 − 2 = 7.
• The identity that removes the difference (§14.1.3, p. 292). The chapter states that taking A but not B is the same as intersecting A with the complement of B. Verified: an outcome survives the left side when it is in A and out of B, and survives the right side under the same two conditions, so the two sides admit the same outcomes.
• Example 1 — one die, primes against odds (§14.1.3, p. 292). S = {1, 2, 3, 4, 5, 6}; A is described as a prime turning up and B as an odd number turning up, giving A = {2, 3, 5} and B = {1, 3, 5}. The chapter asks for four combinations and prints all four: the union is {1, 2, 3, 5}, the intersection is {3, 5}, A but not B is {2}, and the complement of A is {1, 4, 6}. Verified: sizes 4, 2, 1 and 3. Two checks. Verified, not printed: B but not A is {1}, so the two differences are different sets, which is worth showing because the operation is not symmetric.
• Why 1 and 2 are where the mistakes land. Verified: 1 is odd and not prime, which is the sole reason the difference has any member at all on the B side; 2 is prime and not odd, which is the sole reason it has any member on the A side. Both facts are about the numbers, not about probability, and both are load bearing for Example 1's answers.

Figures to have open

• A box labelled S with one shaded region, and with two overlapping shaded regions, redrawn for each operation. Standard schematic. Note that the chapter prints no picture anywhere in §14.1.3 — verified against the page images of pp. 291–292 — so these are not in the book and must not be captioned as the book's.
• A six-by-six grid of ordered pairs for two throws of a die, with the first throw down one axis and the second across the other. An added device; the chapter lists the pairs in running text. This grid is worth building carefully because later topics reuse it.
• A six-cell strip for one die throw, reused four times in section 9.
• The chapter's only numbered figure, Fig 14.1 on p. 301, belongs to the addition-rule topic and is not needed here.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 14 "Probability", §14.1.3 "Algebra of events", pp. 291–292, comprising the four numbered items and Example 1
• The chapter's own pointer back to the earlier chapter on sets, made in the opening sentence of §14.1.3, p. 291 — a deliberate cross-reference out of this chapter, and the only source given for the operations themselves
• The solution to Example 7, p. 304, which applies the complemented-union rule by name and is the chapter's only use of it
• The Summary, p. 312, whose fourth to seventh bullets restate the four operations
• Exercise 14.1 Q2 and Q6, pp. 294–295