NECESSARY AND SUFFICIENT CONDITIONS: QUESTIONS ONLY

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
Work with nondegenerate Euclidean quadrilaterals and the usual inclusive definition of rectangle. Let S mean square and R mean rectangle. Is being a square necessary, sufficient, both or neither for being a rectangle? Draw the arrow and supply a shape that tests the reverse direction.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 2: An even integer
Among integers, is being even necessary, sufficient, both or neither for being divisible by 4? Give an integer showing why the stronger classification fails. Distinguish “divisible by 4” from “equal to 4.”
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 3: A fictional two-part entry rule
For a fictional study room, admission A is permitted if and only if a person has a valid pass P and a booking B. No other entry route exists in this model. Classify P alone, B alone, and the conjunction P and B as conditions for A. Permission is the target, not whether someone actually walks through the door.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 4: An exact scoring definition
A fictional ten-question quiz gives an integer score from 0 to 10. Its complete rule says a pass means a score of at least 8, with no other requirement. Is scoring at least 8 necessary and sufficient for passing? Is scoring exactly 8 necessary? Is scoring exactly 8 sufficient?
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 5: Color and working order
The complete fictional inventory contains four bicycles: a red working bicycle, a red broken bicycle, a blue working bicycle and a blue broken bicycle. Classify being red as a condition for being in working order, within this inventory. Identify a counterexample to each possible one-way guarantee.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 6: If a form is submitted
A fictional processing rule says: “If a form is submitted, a receipt is issued.” Let S mean submitted and R mean receipt issued. Translate this stated rule. Which condition is sufficient for R, and which is necessary for S? Can receipt issuance alone establish submission?
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 7: Released only if approved
An invented archive policy states: “A file is released only if it is approved.” Let L mean released and A mean approved. A draft translation says A → L. Correct it, name the required condition, and explain whether an approved but unreleased file violates the original sentence.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 8: The same words with if
Now the fictional rule is different: “A file is released if it is approved.” Use the same L and A as in case 7. Translate this sentence, compare it with “released only if approved,” and state what must be added to make approval necessary and sufficient for release.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 9: Two alternative signals
A fictional indicator is lit I if and only if either switch R or switch B is on, or both are on. Classify R alone and the inclusive disjunction R or B as conditions for I. Compare this with the two-part admission rule in case 3.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 10: Three observations and a broad claim
You have only three observations of a fictional machine. In two it is ready with its green light on; in one it is not ready with its green light off. Someone claims that a green light is necessary and sufficient for readiness in every future situation. What can these observations establish, and what remains unproved?
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 11: A contrapositive, not a converse
In a fictional record system: “A loan is closed only if its return is logged.” Let C mean loan closed and G mean return logged. Which follows: (a) if the return is not logged, the loan is not closed; (b) if the return is logged, the loan is closed; (c) if the loan is not closed, the return is not logged? Give a possible state that tests the unsupported options.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______

Case 12: Repair a proposed definition
A classmate proposes: “Among positive integers greater than 1, a number is prime if and only if it is odd.” Test both directions separately. Use 2 and 9. Then give a correct defining condition for being prime in this domain.
Target: ______
Condition and arrow: ______
Classification or correction: ______
Reason, counterexample or evidence limit: ______