# Distinguishing definitions from equivalences.

I'm used to people using the symbol $\equiv$ when defining quantities. I might see expressions like

$$\frac{df}{dx} \equiv \lim_{\delta\to 0} \frac{f(x+\delta) - f(x)}{\delta} = 2x \,.$$

where the first part is what is meant by the symbol $df/dx$, whilst the second is an equality that holds in the specific case being considered. On the other hand, what I understand the symbol $\equiv$ to actually mean is 'equivalent to'. A good example of such a usage would be

$$\sin^2 \theta + \cos^2 \theta \equiv 1 \,.$$

The two usages to me seem rather different. The statement above is not a definition of $\sin$ and $\cos$, and certainly not a definition of $1$! Rather it is a statement with content. The first usage of $\equiv$ that I gave is, on the other hand, content-less, for it only serves to define a symbol.

So what I'm wondering is this:

Are the two meanings I've outlined above in fact not all that different? Or if they are, why is it common to use the same symbol for both? Do we need a new symbol for 'defined to be'? I believe I have seen before the symbol

$$\stackrel{\mathrm{def}}{=}$$

but it is used very rarely. It is not possible that the different usages of the symbol could cause confusion among students? Or would most people agree that meaning is clear from context?

Thank you.

I like to use $:=$ to indicate "is defined to be". I believe that this convention is more common in computer science than in math. I personally am very careful about distinguishing non-trivial (i.e. not by definition) identicality $\equiv$ from definition $:=$, but in my experience I seem to be in a very small minority.