How to derive (in)equalities from each other in a notationally sound way?

Question

In elementary algebra and beyond, we are taught to use a sequence of equations to derive a relationship. For instance, to show that $a \le 2b - 1$ follows from $\frac{a+1}{2} = b$, one would use the following sequence of equations, where each equation follows from the previous one.

$$ \begin{align} \frac{a + 1}{2} &= b \newline a + 1 &= 2b \newline a &= 2b-1 \newline a &\le 2b - 1 \end{align} $$

This notation has always seemed inadequate to me. In particular, when I don't want to waste paper I find myself placing multiple equations in one row with an arrow in between them.

$$a = b \Rightarrow c = d $$

This looks nice but does not mean what I want it to mean since in logic

$$a \Rightarrow b \Rightarrow c $$

evaluates to a single value,

$$a \Rightarrow (b \Rightarrow c) $$

whereas I am using it as a short hand for something like

$$ \begin{align} &1. \:& & a \newline &2. \:& & a \Rightarrow b \newline &3. \:& & b \Rightarrow c \newline &4. \:& \therefore \: & \: c \end{align} $$

I've sometimes used the symbol $\rightarrow$ as in

$$a = b \rightarrow c = d $$

but what I really want is to chain implications so that I am emphasizing the nature of my derivation as being a logical progression.

I've also used $\equiv$ as in

$$a = b \equiv c = d $$

but this does not work in many situations such as deriving $a \le 2b - 1$ from $a = 2b - 1$.

Am I overthinking this? Should I just use the typical chain of equations with no symbol to represent the relationship between those equations? Are any of the aforementioned shorthand notations appropriate?

Thanks

Answer 1

This is a soft-answer.

I think (from your choice of the tags propositional-logic and predicate-logic, and from your remark " in logic ...evaluates to a single value") that you are not so much overthinking this as confused about the use of the symbol $\implies$ when we are writing in the mathematical dialect of English. It is not a logical symbol in the whatever-calculus, it is just a convenient abbreviation for "which implies" or "which in turn implies".

So the way we should read

$P\implies Q \implies R$

is

"$P$ implies $Q$, which in turn implies $R$".

Answer 2

What you are using repeatedly is the transitivity of implication, that is :

" IF $( A\rightarrow B)\land (B\rightarrow C)$ THEN $(A\rightarrow C)$".

Solving an equation/ inequation can be seen as a reasoning :

(1) If the solution set of this expression is not empty, then ... then ... then the solution set has to be ...

(2) A) The solution set I have found works : I have solved the equation / inequation

(2) B) The solution set I have found does not work , there is no solution, since my reasoning showed the solution set to which I arrived was the only possible one.

What I mean is that the chain of implications only allows to arrive at a necessary condition as to the solution set. For we are working under the hypothesis that the equation / inequation can be solved.

Reference : Richarson , Fundamentals of mathematics ( at available at archive.org)