Correct Option (3)
Statements 1 ("All trains are run by diesel engine") and 3 ("No train is run by diesel engine") are contrary propositions. According to the classical square of opposition, contrary statements cannot both be true simultaneously. If all trains are run by diesel, it is logically impossible for no train to be run by diesel. Conversely, if no train is run by diesel, it is impossible for all trains to be run by diesel. However, both statements can be false. This occurs in a scenario where some trains are run by diesel engines, and some are not (e.g., electric trains also exist). In such a case, neither Statement 1 nor Statement 3 would be true, thus making both false. This precisely matches the condition: "two cannot both be true, but both can be false."
Incorrect Options:
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1 & 2: Statement 1 ("All trains are run by diesel engine") implies Statement 2 ("Some trains are run by diesel engine") under the assumption of existential import. Therefore, if Statement 1 is true, Statement 2 must also be true. They can both be true, which violates the condition "cannot both be true."
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2 & 3: Statement 2 ("Some trains are run by diesel engine") and Statement 3 ("No train is run by diesel engine") are contradictory propositions. Contradictory statements cannot both be true, and they also cannot both be false. If one is true, the other must be false, and vice versa. This violates the condition "both can be false."
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1 & 4: Statement 1 ("All trains are run by diesel engine") and Statement 4 ("Some trains are not run by diesel engine") are contradictory propositions. Similar to the pair 2 & 3, contradictory statements cannot both be true and cannot both be false. This violates the condition "both can be false."