Exercise 93 - Chapter 1

Fredrick Eisele

Complete the proof

1. Show, that if f is left adjoint to g then for any q ∈ Q, we have f (g(q)) ≤ q.
2. 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 $$ \text{For every } p \in P \:\&\: q \in Q \\ p \le g(f(p)) \:\&\: f(g(q)) \le q $$

David Lambert

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.

1. 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.