GATE CS 2006 Q26 — Propositional and First Order Logic

To translate the English statement into first-order logic, we can break it down into smaller logical components:

1. "Tigers and lions...": This refers to any animal $x$ that is either a tiger or a lion. In natural language, "and" is used here to group two classes together, meaning the statement applies to any member of either class. Thus, the antecedent is: $$\text{tiger}(x) \lor \text{lion}(x)$$ Note: Using $\text{tiger}(x) \land \text{lion}(x)$ (as in Option A) would incorrectly imply an animal that is simultaneously both a tiger and a lion.
2. "...attack if they are hungry or threatened.": The phrase "$P$ if $Q$" translates to the conditional statement $Q \rightarrow P$. Here, the condition $Q$ is "hungry or threatened" and the consequence $P$ is "attacks". This translates to: $$(\text{hungry}(x) \lor \text{threatened}(x)) \rightarrow \text{attacks}(x)$$

Combining these two parts for all individuals $x$ in the domain, we get:

This matches Option D.

Why other options are incorrect:

• Option A: Uses $\text{tiger}(x) \land \text{lion}(x)$, which means the statement only applies to creatures that are both a tiger and a lion at the same time.
• Option B: Asserts that every tiger or lion is always hungry or threatened, and always attacks.
• Option C: Represents the converse: "if they attack, then they are hungry or threatened" (i.e., $\text{attacks}(x) \rightarrow (\text{hungry}(x) \lor \text{threatened}(x))$).