NECESSARY AND SUFFICIENT CONDITIONS: EXPLAINED ANSWER KEY

All institutional rules, inventories and machine observations below are invented. Mathematical cases use their stated domains. For each question, keep the target fixed, write the arrow and give a reason or counterexample. Use classical propositional logic for the explicit rule translations; inclusive or allows both alternatives. Do not infer a causal mechanism or a real-world guarantee from a classroom stipulation.

Case 1: Squares and rectangles
Sufficient, but not necessary: S → R. A rectangle need not be a square.
A square has four right angles, so every square meets the rectangle definition. A rectangle with side lengths 2 and 5 is a counterexample to R → S: it is a rectangle but its sides are not all equal. The counterexample does not challenge S → R. Do not use the everyday contrast “square versus rectangle” to exclude squares from rectangles after the question has explicitly chosen the inclusive mathematical definition. Relative to the same domain, being a rectangle is necessary for being a square. The two descriptions name the same one-way connection from different ends.

Case 2: An even integer
Necessary, but not sufficient. Divisible by 4 → even; even does not imply divisible by 4.
If an integer n is divisible by 4, n = 4k for some integer k. It is therefore 2 times the integer 2k, so it is even. The integer 6 is even but is not divisible by 4, disproving the reverse implication. The integer 8 also makes clear that the condition concerns divisibility, not equality to one number. This argument works for zero and negative integers as well: zero is divisible by 4, and −8 is divisible by 4 and even. No restriction to positive integers is needed for this particular result.

Case 3: A fictional two-part entry rule
P and B are each necessary but not sufficient. Together, P ∧ B is necessary and sufficient for A.
The exact rule is A ↔ (P ∧ B). A permitted entrant must satisfy both checks. A pass holder with no booking fails the full condition, and a booked person without a pass also fails. When both checks hold, the stated rule permits admission; no unstated third requirement is part of this model. Passing just one check does not finish the task. Conversely, “permitted” does not imply that the person chooses to enter. Keeping the target precise avoids converting an eligibility rule into a prediction about behavior. This rule is invented for the exercise and describes no actual institution.

Case 4: An exact scoring definition
At least 8 is necessary and sufficient. Exactly 8 is sufficient, but not necessary.
Under the supplied definition, passing and having a score of 8, 9 or 10 coincide. Write Pass ↔ (score ≥ 8). A score of exactly 8 lies inside that range and therefore guarantees a pass, but scores of 9 and 10 are counterexamples to the claim that exactly 8 is required. The difference between “at least” and “exactly” changes the condition even though the numeral stays the same. No claim about a real exam, extra attendance requirements or rounding rules is being made. If a real course adds another requirement, its pass condition must be checked again rather than borrowed from this model.

Case 5: Color and working order
Neither necessary nor sufficient within the stated four-bicycle inventory.
The blue working bicycle disproves the claim that red is necessary for working order: the target occurs without red. The red broken bicycle disproves the claim that red is sufficient: red occurs without the target. Name both witnesses because one counterexample cannot do both jobs here. The working red bicycle shows overlap, which is entirely compatible with “neither.” The broken blue bicycle adds no guarantee. This is an exhaustive invented inventory, so the classification concerns these four bicycles. It does not describe manufacturing quality, all red bicycles or whether paint affects mechanical performance.

Case 6: If a form is submitted
S → R. Submission is sufficient for receipt issuance; receipt issuance is necessary for submission under this rule. R alone does not establish S.
The conditional forbids a submitted form without a receipt. It does not forbid a receipt issued for another reason. A situation with R true and S false satisfies the stated one-way rule and defeats the proposed converse R → S. In this exercise, the rule is treated as a logical constraint over the relevant situations. We are not testing a real software service or using the truth table as evidence of reliable delivery. A receipt may be an effect of submission in a real system, but the displayed logical statement by itself neither supplies that mechanism nor proves that a particular submission occurred.

Case 7: Released only if approved
The correction is L → A. Approval is necessary for release. An approved but unreleased file does not violate this one-way rule.
“Only if approved” attaches a requirement to release: no release without approval. It does not instruct the archive to release every approved file. The proposed A → L instead makes approval sufficient for release and forbids an approved file waiting in the queue, which is stronger than the given sentence. Under the original rule, a released and unapproved file is the forbidden case. An approved and unreleased file is allowed. Do not infer “approval is not sufficient in every real archive”; the narrower conclusion is that this sentence alone does not make it sufficient. Other policies could add the missing direction.

Case 8: The same words with if
“Released if approved” is A → L. To get both directions, also require L → A, giving L ↔ A.
In the classroom reading, “L if A” reverses the order of the clauses: A is the sufficient condition for L. It does not rule out a release under another authorization route. “L only if A” instead rules out release without approval. Combining the two gives an exact equivalence between approval and release in the model. Do not replace “if” with “if and only if” merely because a real policy probably intended a complete procedure. Ask for clarification before strengthening a real requirement. This pair of cases changes the sentence rather than pretending that the two rules already said the same thing.

Case 9: Two alternative signals
R alone is sufficient but not necessary. R ∨ B is necessary and sufficient for I.
The model is I ↔ (R ∨ B). Turning R on meets the stated condition, even if B is off. But with R off and B on, I is still lit, so R is not necessary. The whole disjunction is necessary because the indicator is not lit when both switches are off; it is sufficient because at least one on-switch makes the condition true. Inclusive “or” allows both switches to be on. In case 3, each component was required and neither alone sufficed. In this case, either component alone suffices and neither is required. The connector changes the answer; the number of named components does not.

Case 10: Three observations and a broad claim
Green and ready coincide in these three observations. The universal claim about all future situations is not established by this evidence.
Within the observed rows, neither mismatch occurs. But the supplied evidence leaves unobserved situations open: a green light with the machine not ready would refute sufficiency, and readiness with the light off would refute necessity. Proposing those possibilities does not mean either fault actually occurred. It identifies the evidence needed to test the guarantee. A complete, reliable specification might justify a stronger conclusion, while an additional contrary observation could disprove one direction. “Not established” describes an evidence limit, not a finding that the universal relation is false. Keep the scope of the answer at the scope of the information supplied.

Case 11: A contrapositive, not a converse
Only (a) follows from C → G. It is ¬G → ¬C, the contrapositive. Options (b) and (c) require an extra direction.
A closed but unlogged loan would violate C → G. Consequently, lack of a log rules out closure when the rule is assumed true. A logged return with the loan still open satisfies the original rule but makes (b) false and (c) false. That one state has G true and C false. Notice that the contrapositive both reverses the arrow and negates both ends. Simply reversing the arrow gives the converse G → C, and simply negating both ends gives the inverse ¬C → ¬G. They are equivalent to each other, but neither is guaranteed by C → G alone.

Case 12: Repair a proposed definition
Oddness is neither necessary nor sufficient for primality. A prime has exactly two positive divisors: 1 and itself.
The number 2 is prime and is not odd, so it refutes Prime → Odd and defeats the necessity claim. The number 9 is odd but has positive divisors 1, 3 and 9, so it refutes Odd → Prime and defeats the sufficiency claim. A positive integer greater than 1 is prime exactly when its only positive divisors are 1 and itself. Naming 2 alone repairs neither direction: it diagnoses one failure. Naming 9 alone diagnoses the other. An “if and only if” definition requires both tests, even when one half seems plausible or agrees with several familiar examples.

References:
https://plato.stanford.edu/entries/necessary-sufficient/
https://openstax.org/books/introduction-philosophy/pages/5-3-arguments