Algebraic equalities and their respective (dis)proofs: Distribution law

Continuing on this small project regarding proofs (see “Aleksey” 2022), I will be showing off algebraic equalities and working out whether-or-not they are just conjecture or if they are bona fide theorems or lemma. The Algebra and Calculus textbook written by E. Y. Amiran (2014, pp. 23–-25) gives students of mathematics algebraic equality problems to practice their proving skills. In this article, I will be focusing on a subset of these problems that make use of the distributive property.

Contents at a glance

1. Background
2. Case studies
3. End matter
4. References

Background

I started off this series introducing the notion of an algebraic equality (“Aleksey” 2022) and I recommend reading that first before proceeding with this article. Students of elementary algebra are taught the distributive property applied to real numbers (OpenStax 2020, §1.5); I will introduce the following generalised form of the law:

This little-known fact can be used as a justification for all kinds of algebraic equalities. Furthermore, I will continue to use JupyterLab desktop app (Bektas 2021) to automate the process of equation graphing. I made a slight change to the core logic of the plotting utility:

def plot_functions(f_l, f_r, r_l=[0, 100, 0.1], r_r=[0, 100, 0.1]):
    # [... snip ...]…