Two statements are given, followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
* All roads are routes.
* All routes are bridges.
Conclusions:
* I. Some routes are roads.
* II. All roads are bridges.
- ANeither conclusion I nor II follows
- BBoth conclusions I and II follow
- COnly conclusion II follows
- DOnly conclusion I follows
Solution & Step-by-step Explanation
Let's analyze the statements using Venn diagrams or logical containment relations:
* Statement 1: "All roads are routes" implies that the set of 'Roads' is completely contained inside the set of 'Routes' ().
* Statement 2: "All routes are bridges" implies that the set of 'Routes' is completely contained inside the set of 'Bridges' ().
Combining both statements gives a nested relation:
Now, let's evaluate the given conclusions:
* Conclusion I: "Some routes are roads."
Since all roads are inside routes, the portion of routes occupied by roads ensures that some routes are definitely roads. Thus, conclusion I is true and follows.
* Conclusion II: "All roads are bridges."
Since the set of 'Roads' is inside 'Routes', and 'Routes' is entirely inside 'Bridges', the set of 'Roads' is automatically inside 'Bridges' (). Thus, conclusion II is also true and follows.
Therefore, both conclusions I and II follow.
* Statement 1: "All roads are routes" implies that the set of 'Roads' is completely contained inside the set of 'Routes' ().
* Statement 2: "All routes are bridges" implies that the set of 'Routes' is completely contained inside the set of 'Bridges' ().
Combining both statements gives a nested relation:
Now, let's evaluate the given conclusions:
* Conclusion I: "Some routes are roads."
Since all roads are inside routes, the portion of routes occupied by roads ensures that some routes are definitely roads. Thus, conclusion I is true and follows.
* Conclusion II: "All roads are bridges."
Since the set of 'Roads' is inside 'Routes', and 'Routes' is entirely inside 'Bridges', the set of 'Roads' is automatically inside 'Bridges' (). Thus, conclusion II is also true and follows.
Therefore, both conclusions I and II follow.