Imagine there's a set named \( S \) which consists of integer elements, and this set is infinite. From \( S \), two subsets are defined: \( L \) consists of all negative integers in \( S \), and \( G \) contains all non-negative integers from \( S \). You need to consider two propositions, \( P \) and \( Q \). Proposition \( P \) states that subset \( G \) is infinite, and proposition \( Q \) suggests that subset \( L \) has a finite number of elements. Given these conditions, you should: (a) Provide an example of a set \( S \) that makes proposition \( P \) true (meaning \( G \) is indeed infinite) and proposition \( Q \) false (meaning \( L \) is not finite). If there's no such set, explain why it's not possible. (b) Give an example of a set \( S \) that makes proposition \( P \) false (meaning \( G \) is not infinite) and proposition \( Q \) true (meaning \( L \) is finite). If this cannot be done, explain the reason. (c) Clarify the logical relationship between proposition \( P \) and \( Q \). Are they equivalent? Does one imply the negation of the other, or are they unrelated? A concise answer is enough, without further explanation needed.
Imagine there's a set named \( S \) which consists of integer elements, and this set is infinite. From \( S \), two subsets are defined: \( L \) consists of all negative integers in \( S \), and \( G \) contains all non-negative integers from \( S \).
You need to consider two propositions, \( P \) and \( Q \). Proposition \( P \) states that subset \( G \) is infinite, and proposition \( Q \) suggests that subset \( L \) has a finite number of elements.
Given these conditions, you should:
(a) Provide an example of a set \( S \) that makes proposition \( P \) true (meaning \( G \) is indeed infinite) and proposition \( Q \) false (meaning \( L \) is not finite). If there's no such set, explain why it's not possible.
(b) Give an example of a set \( S \) that makes proposition \( P \) false (meaning \( G \) is not infinite) and proposition \( Q \) true (meaning \( L \) is finite). If this cannot be done, explain the reason.
(c) Clarify the logical relationship between proposition \( P \) and \( Q \). Are they equivalent? Does one imply the negation of the other, or are they unrelated? A concise answer is enough, without further explanation needed.
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