Counting false entries in truth tables of bracketed formulae connected by implication

TL;DR

This paper analyzes the number of false entries in truth tables of all bracketed implications with n variables, providing a recurrence, asymptotic formula, and ratio convergence.

Contribution

It introduces a recurrence relation and asymptotic formula for counting false entries in truth tables of bracketed implications, a novel combinatorial analysis.

Findings

Derived a recurrence relation for f_n

Established an asymptotic formula for f_n

Proved the ratio of false entries converges to (3 - sqrt(3))/6

Abstract

In this paper we count the number of rows f_n with the value "false" in the truth tables of all bracketed formulae with n distinct variables connected by the binary connective of implication. We find a recurrence and an asymptotic formulae for f_n. We also show that the ratio of f_n to the total number of rows converges to \frac{3-\sqrt{3}}{6}.