Subjects logic

Logic Translation

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Logic Translation


1. The problem asks to translate logical statements into English, where $R(x)$ means "$x$ is a rabbit" and $H(x)$ means "$x$ hops", with the domain being all animals. 2. For statement (a) $\forall x (R(x) \to H(x))$: This means "For every animal $x$, if $x$ is a rabbit, then $x$ hops." In other words, all rabbits hop. 3. For statement (b) $\forall x (R(x) \wedge H(x))$: This means "For every animal $x$, $x$ is a rabbit and $x$ hops." In other words, every animal is a rabbit and hops. 4. For statement (c) $\exists x (R(x) \to H(x))$: This means "There exists at least one animal $x$ such that if $x$ is a rabbit, then $x$ hops." This is logically weaker than (a) because it only requires one animal for which the implication holds. 5. For statement (d) $\exists x (R(x) \wedge H(x))$: This means "There exists at least one animal $x$ such that $x$ is a rabbit and $x$ hops." In other words, there is at least one hopping rabbit. Final answers: (a) All rabbits hop. (b) Every animal is a hopping rabbit. (c) There is at least one animal for which if it is a rabbit, it hops. (d) There is at least one hopping rabbit.