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Options

Complete the proof

- Show, that if f is left adjoint to g then for any q ∈ Q, we have f (g(q)) ≤ q.
- Show that If 1.6 holds, then for any p ∈ P and q ∈ Q, if p ≤ 1(q) then f (p) ≤ q.

**Galois connection**
$$
f(p) \le q \iff p \le g(q) \\
\text{Where g is right-adjoint to f and f is left-adjoint to g}
$$
**1.6**
$$
\text{For every } p \in P \:\&\: q \in Q \\
p \le g(f(p)) \:\&\: f(g(q)) \le q
$$

## Comments

Since the first inequality holds by reflexivity, the desired relation must also hold.

`1. By adjointness of f,g: g(q)\\(\leq\\)g(q)\\(\\Leftrightarrow\\)f(g(q))\\(\leq\\)q. Since the first inequality holds by reflexivity, the desired relation must also hold. 2. Assume p≤g(q). By monotonicity of f, f(p)≤f(g(q)). By 1.6 and transitivity, f(p)≤q, as was to be proved.`