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We know that if A=B and B=C, then A=C [Law of transitivity].

The same thing should be applicable for proportions right? i.e. if A is proportional to B, and B is proportional to C, then A should be proportional to C?

This should be logical right?

But in a lot of equations without fail, I find that A is inversly proportional to C

For example, (I have used an elementary physics equation, but the question is about mathematics)

In Ohm's law, we know that:

V= IR

Which implies:

a) V ∝ I

b) I ∝ 1/R

c) V ∝ R

Here from a and b we should imply that V ∝ 1/R, which directly contradicts c.

Whenever three variables are related in this fashion, does it always imply one thing or the other, or do they have no particular relation?

I know that R is just a proportionality constant, but does this make it incorrect?

What is going wrong here? Am I missing something?

batchcoding____s
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1 Answers1

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From the equation

$$V=IR$$

the statement "$V$ is proportional to $I$" means that all else held equal, $V$ is a fixed multiple of $I$ (or, rephrased, the ratio between $V$ and $I$ is a constant number).

With this sense of the words "proportional to", the relation is not transitive; indeed, you have a counterexample already.

So in what sense is the relation "proportional to" transitive? Let's define it such:

"A variable $X$ is proportional to another variable $Y$ when there is a nonzero constant number $k$ such that the equation $X=kY$ always holds true."

That definition of proportionality is transitive; indeed, let $X$, $Y$, $Z$ be variables such that $X$ is proportional to $Y$ and $Y$ is proportional to $Z$. So there are nonzero constants $k_1$ and $k_2$ such that the equations $X=k_1Y$ and $Y=k_2Z$ always holds true. Substituting yields

$$X=(k_1k_2)Z$$

and therefore $X$ and $Z$ are proportional to each other.

user_194421
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