Introduction / Context:
This item checks syllogistic reasoning across sets using three premises: all doors are roads, no road is a fruit, and some flowers are doors. You must decide which conclusions necessarily follow, which are impossible, and which are indeterminate. Paying attention to subset relations and universal negatives is essential.
Given Data / Assumptions:
- D ⊆ R (all doors are roads)
- R ∩ F = ∅ (no road is a fruit)
- ∃ element that is both a flower and a door (Some flowers are doors)
Concept / Approach:
Use set inclusion and disjointness. From D ⊆ R and R ∩ F = ∅, any door cannot be a fruit. From Some flowers are doors and D ⊆ R, those particular flowers are also roads, and thus not fruits. Be careful: a statement about some flowers does not automatically tell you about all flowers.
Step-by-Step Solution:
Test III: Some roads are flowers. Since some flowers are doors and all doors are roads, at least those flowers are also roads. Therefore, III is true.Test I: Some fruits are doors. Impossible because doors are roads and no road is a fruit. So I is false.Test II: Some fruits are flowers. Undetermined. The flowers that are doors cannot be fruits, but there might be other flowers that are fruits. Nothing forbids this.Test IV: No fruit is a flower. Also undetermined. The premises do not prohibit non-road flowers from being fruits, so asserting none is too strong.
Verification / Alternative check:
Construct a Venn-style scenario: let some flowers be doors (and hence roads, not fruits). Add other flowers outside roads; these could either overlap fruits or not. In both scenarios III holds; II and IV are mutually exclusive possibilities.
Why Other Options Are Wrong:
A: Claims IV follows; it does not.C: Includes I, which contradicts the premises.D: Omits III, which certainly follows.E: Says all follow; I is impossible and II/IV cannot both hold.
Common Pitfalls:
Assuming properties of all flowers from a statement about some flowers; forgetting that a universal negative about roads does not say anything about non-road flowers.
Final Answer:
Only either II or IV, and III follow
Discussion & Comments