1. The problem defines the set $S = \{3x - 1 \mid x \in \mathbb{N} \wedge 5 \leq x \leq 8\}$. 2. This means $S$ contains values of the expression $3x - 1$ where $x$ is a natural number between 5 and 8 inclusive. 3. We will evaluate $3x - 1$ for each $x$ in $\{5,6,7,8\}$. 4. For $x=5$: $$3(5) - 1 = 15 - 1 = 14$$ 5. For $x=6$: $$3(6) - 1 = 18 - 1 = 17$$ 6. For $x=7$: $$3(7) - 1 = 21 - 1 = 20$$ 7. For $x=8$: $$3(8) - 1 = 24 - 1 = 23$$ 8. Therefore, the set $S$ is $$S = \{14, 17, 20, 23\}$$. This completes the evaluation of the set $S$.