Among the following propositions two are related in such a way that they cannot both be true but can both be false. Select the code that states those two propositions. Propositions: (a) Every student is attentive. (b) Some students are attentive. (c) Students are never attentive. (d) Some students are not attentive. Codes:
(a) and (c)
To solve the question of which propositions cannot both be true but can both be false, we need to understand the nature of the propositions provided. They belong to a category of logical statements known as categorical propositions.
- The proposition "(a) Every student is attentive" is a universal affirmative statement. It asserts that all members of the group 'students' have the attribute 'attentive'.
- The proposition "(b) Some students are attentive" is a particular affirmative statement. It claims that at least one member of the group 'students' is 'attentive'.
- The proposition "(c) Students are never attentive" is a universal negative statement. It denies that any member of the group 'students' has the attribute 'attentive'.
- The proposition "(d) Some students are not attentive" is a particular negative statement. It asserts that at least one member of the group 'students' lacks the attribute 'attentive'.
Now, we need to identify the pair of propositions where both cannot be true at the same time, yet can both be false:
- Propositions (a) and (b) can both be true simultaneously, as both indicate the presence of at least some attentive students.
- Propositions (b) and (c) cannot both be true, as (b) requires some attentiveness and (c) denies any attentiveness, but can both be false.
- Propositions (a) and (c) are directly contradictory because one asserts that every student is attentive, while the other asserts that no student is attentive. Therefore, they cannot both be true simultaneously but can be false if some students are attentive, while others aren't (satisfying proposition d).
- Propositions (c) and (d) can both be true and false, as (c) argues a universal absence of attentiveness, which could coexist with some students being inattentive as in (d).
Thus, the correct answer is the pair of propositions where both cannot be true but can both be false: (a) and (c).
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