Four True Statements About the Book Club
Twelve students belong to the Thornfield Book Club. A survey records four statements about their winter reading, and every statement is true: (1) Every member who read the mystery also read the biography. (2) Some members who read the biography did not read the poetry. (3) No member who read the poetry read the mystery. (4) Exactly three members read the poetry, and exactly seven read the biography. Ms. Okafor asks the club to state only what the four statements force. Three members volunteer conclusions. Jonah says: 'Exactly four members read the biography but not the poetry.' Lena says: 'At least one member read the biography but not the mystery.' Rafa says: 'If exactly five members read the mystery, then at least one poetry reader did not read the biography.'
1. For each of the three conclusions, say whether it is forced by the four statements, and justify each verdict.
Read the worked answer
Jonah - not forced. Biography = 7 and poetry = 3, with 12 members, so the overlap can run from 0 (7 + 3 = 10 ≤ 12) up to 3. 'Biography but not poetry' therefore ranges from 4 to 7. Jonah's 'exactly four' is only the smallest case.
Lena - not forced. Nothing stops the mystery readers from being exactly the 7 biography readers. If they are, every biography reader read the mystery, so nobody read biography-without-mystery. Statements (1) and (3) are satisfied, and (2) is satisfied because those 7 read no poetry. Her conclusion could be true, but the statements do not force it.
Rafa - forced. Suppose 5 read the mystery. By (1) those 5 are biography readers; by (3) none of them read poetry. So at least 5 of the 7 biography readers read no poetry, leaving at most 2 biography readers inside the poetry group of 3. At least one poetry reader must therefore lie outside the biography group.
2. What is the largest possible number of mystery readers, and what is the smallest? Show that each bound can actually be reached, not merely that it is an upper or lower limit.
Read the worked answer
From (1), mystery readers are a subset of the 7 biography readers. From (3), no mystery reader is a poetry reader.
Maximum = 7. Let all 7 biography readers read the mystery, and put the 3 poetry readers entirely outside the biography group. Check: 7 + 3 = 10 ≤ 12, so 2 members read nothing. (1) holds, (3) holds (no overlap), (2) holds because all 7 biography readers read no poetry. So 7 is achieved, not just a ceiling.
Minimum = 0. Let 3 members read poetry only, 7 read biography only, 2 read nothing. (1) is vacuously true, (2) holds, (3) holds because the poetry and mystery groups are both empty of overlap. So 0 is achieved as well.
3. Ms. Okafor wants Jonah's conclusion to become forced. Give one additional true statement about the club that would do it, and explain why it works.
Read the worked answer
Add: 'All three poetry readers also read the biography.'
Then the biography-and-poetry overlap is exactly 3, so biography-but-not-poetry = 7 − 3 = 4, and Jonah's 'exactly four' is forced.
Why one number is enough: the biography total (7) and poetry total (3) are already known, so fixing the overlap fixes every other cell of the two-set table. Note the asymmetry with Rafa's conclusion - a statement about the overlap pins the count, whereas a statement about the mystery total only pins it once the overlap is bounded.