By Pathantu · Published 6 October 2026 · Original practice, not PYQ
Method
All A are B: draw A inside B. No A is B: disjoint sets. Some A are B: at least one shared member.
Do not reverse “All A are B”. You may reverse “Some A are B” because the shared member belongs to both.
Try to satisfy all statements while making the conclusion false. One valid counterexample disproves certainty.
Six questions: try first, then open the solution
1. All pens are tools; all tools are objects. Are all pens objects?
Show worked solution
Yes: pens ⊆ tools ⊆ objects. Inclusion carries through both sets.
2. Some books are blue; all blue things are coloured. Are some books coloured?
Show worked solution
Yes. The blue books exist and are coloured.
3. All roses are flowers; some flowers are red. Must some roses be red?
Show worked solution
No. All the red flowers might lie outside the rose set.
4. No cats are birds; some pets are cats. Must some pets not be birds?
Show worked solution
Yes. The pets that are cats cannot be birds.
5. All A are B. Does this prove all B are A?
Show worked solution
No. A={1}, B={1,2} satisfies the statement but 2 is outside A.
6. Some A are B; some B are C. Must some A be C?
Show worked solution
No. A={1}, B={1,2}, C={2} meets both statements, yet A and C do not overlap.
Check your reasoning
“Does not follow” is not the same as “impossible”. Check any different convention stated in a test.
For each wrong answer, write the incorrect step and solve the question again without the solution. These open lessons need no login; timed quizzes require a student account.
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