Question 42

The relation $\preccurlyeq$ is the pointwise product order on $\{0, 1\}^n$ (isomorphic to $(\mathcal{P}(\{1, \dots, n\}), \subseteq)$):

• Reflexive: $f(x) \le f(x)$ for all $x$, so $f \preccurlyeq f$.
• Antisymmetric: If $f(x) \le g(x)$ and $g(x) \le f(x)$ for all $x$, then $f = g$. (Hence $\preccurlyeq$ is not symmetric, ruling out A and D).
• Transitive: If $f \preccurlyeq g$ and $g \preccurlyeq h$, then $f(x) \le g(x) \le h(x)$ for all $x$, so $f \preccurlyeq h$.
  Thus $(\mathcal{F}, \preccurlyeq)$ is a partial order (B).
• Moreover, every pair $f, g \in \mathcal{F}$ has a greatest lower bound $\text{glb}(f, g)(x) = \min(f(x), g(x))$ and a least upper bound $\text{lub}(f, g)(x) = \max(f(x), g(x))$ in $\mathcal{F}$, so $(\mathcal{F}, \preccurlyeq)$ is a lattice (C).