Q.Show that $\mathbb{N} \subseteq \mathbb{W}$ and $\mathbb{N} \subset \mathbb{W}$, where $\mathbb{N} = \{1, 2, 3, \dots\}$ and $\mathbb{W} = \{0, 1, 2, 3, \dots\}$.

Check whether every natural number is also a whole number, and then check whether the two sets are actually equal or genuinely different.

Step 1 — Test $\mathbb{N} \subseteq \mathbb{W}$. Take any $x \in \mathbb{N}$, so $x \in \{1, 2, 3, \dots\}$. Every such $x$ is also a member of $\mathbb{W} = \{0, 1, 2, 3, \dots\}$, since $\mathbb{W}$ contains every natural number plus the extra element $0$. So every element of $\mathbb{N}$ is an element of $\mathbb{W}$, which is exactly the definition of $\mathbb{N} \subseteq \mathbb{W}$.

Step 2 — Check whether $\mathbb{N} = \mathbb{W}$. Consider $0$: $0 \in \mathbb{W}$, but $0 \notin \mathbb{N}$ (since $\mathbb{N}$ starts at $1$). So $\mathbb{W}$ contains an element $\mathbb{N}$ does not — the two sets are not equal. …