Question

Every Girl Scout who sells at least 30 boxes of cookies will get a prize. Suzy, a Girl Scout, got a prize. Therefore. Suzy sold at least 30 boxes of cookies. Part 2. Determine whether each argument is valid. If the argument is valid, give a proof using the laws of logic. If the argument is invalid, give values for the predicates P and Q over the domain a,b that demonstrate the argument is invalid. (3) \begin{tabular} ∃x(P(x)∧Q(x)) \\ ∴∃xQ(x)∧∃xP(x) \end{tabular} (b) \begin{tabular} \hline∀x(P(x)∨Q(x)) \\ ∴∀xQ(x)∨∀xP(x) \\ \hline \end{tabular}

Answer 1

Using principles of logical inference, the first argument concerning Existential Instantiation is valid, while the second argument concerning Distribution is invalid.

Step-by-step explanation:

In the field of Mathematics, specifically in Logic, the given arguments can be evaluated for validity using principles of logical inference. The first argument is a classic instance of Existential Instantiation, while the second is an example of Distribution.

Existential Instantiation: If it's stated that there exists an x such that P(x) and Q(x) are true (∃x(P(x)∧Q(x))), it necessarily implies that there exists an x such that P(x) is true and there exists an x such that Q(x) is true (∃xP(x)∧∃xQ(x)). Hence, the first argument is valid.

Distribution: The second argument states that for all x, either P(x) or Q(x) is true. (i.e., ∀x(P(x)∨Q(x))). However, this does not imply that for all x, P(x) is true, or for all x, Q(x) is true. (i.e., it is not equivalent to ∀xP(x)∨∀xQ(x)). Hence, the second argument is invalid.

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