# Section 1.3: Problem 3 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Carry out the argument for Lemma 13B for the case of the operation $\mathcal{E}_{\neg}$ .
Given $\alpha\in S$ , the proper initial segments of $\mathcal{E}_{\neg}(\alpha)=(\neg\alpha)$ are $($ , $(\neg$ , $(\neg\alpha_{0}$ (where $\alpha_{0}$ is a proper initial segment of $\alpha$ if one exists), $(\neg\alpha$ . Given that by the induction hypothesis $\alpha_{0}$ has more left parentheses than right ones, and that $\alpha$ has a balance of left and right parentheses, we conclude that all proper initial segments of $\mathcal{E}_{\neg}(\alpha)$ have more left parentheses than right ones.