I'm aware of the idea of $:=$ as 'definition, similar to an equality', but I'm confused about how it's used, and whether we mean to define the expression's value for all conditions. What do we mean by definition?
If I define $y:=x^2,$ this makes sense, but what about $$x^2+y^2:=1\;?$$ What am I defining here? How does my 'definition' mean differently from simply stating an equality?
How about $$\sin(x)^2+\cos(x)^2:=1\; ?$$ This always has the value of $1$ for any $x$ so how do I 'define' anything on this, it's either true or false, it's not my definition that makes it true, it is true always.
How do we find truth values for statements given with '$:=$', and what if I make a false equality for all $x$ like $$3x:=x+x$$ how do I 'define' something that can't be true?
ADDENDUM
I'm slightly struggling with these replies, because of my lack of understanding of what 'definition' means.
Writing $x^2+y^2:=1$, which you may technically do, is defining the entire symbol $x^2+y^2$ to be 1. The individual parts of the symbol, $x^2$, + and $y^2$ might have no additional meaning.
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You are defining something to be something else. If I define a twinkie to be something, any instance of the twinkie is that something. The cream filling or any other knowledge of a similar looking object mean nothing. I could have replaced the $x^2+y^2$ with a smiley face with $x^2$ and $y^2$ substituted for its eyes. It's just a symbol.
What does this mean in terms of 'definition'? Surely we can define a relation (for example the unit circle) without having to somehow verge away from the expression having it's usual mathematical meaning?
Is it ok to make an operation over some definition at the very first time you are defining it?
This question seems to suggest that for some people, using the definition symbol suggests that what is on the left feels like 'some symbol' instead of the mathematical expression it should define.
Best Answer
Then let's go back to definitions (always a good place to start): a definition is a specification of meaning; a mathematical object may be defined by assigning properties to it, by equating it with another object, etc.
A literal interpretation of this definition is that it is legitimising replacing every instance of $\text“x+x\text”$ with $3x$ (here, $\text“3x”$ is the subject of the definition).
It could also mean that in the given system, given an input $x,$ the operations $3x$ and $x+x$ return the same result. (There might be an earlier definition giving how the operation $x+x$ works.)
If, additionally, you are working in an arithmetical system, then this definition will lead to inconsistencies; not because $$3x=x+x\tag2$$ is false equality, but precisely because it is (forced to be) true!
The equality $(2)$ is—by definition $(1)$— true. When given a definition, within its system, it is true by assumption.
The statement of a definition is taken as true.
Sometimes, a definition leads to inconsistencies (for example, the statement of the definition turning out to be also false); other times, they are ambiguous, in which case we say that the object being defined is not well-defined.
The point is that creating definitions is a design process, and some designs are shoddy, because issues that crop up are not dealt with, or potential issues are not noticed.
And like all good designs, good definitions commonly undergo multiple revisions: a working definition may have no problem, but I may still keep refining it such that my work flows more naturally, and the final definition might differ drastically from the preliminary one.
You are defining your points $(x,y)$ of interest precisely as the set of points on the unit circle centred at the origin.
A definition is a starting point for further work. It may then be invoked in the derivation of a result/theorem or answer.
In the above, I have used your given definition as a starting point in characterising your points of interest as the unit circle centred at the origin.
Please do not conflate a definition and a result/theorem, or a definition and a characterisation.
It's up to the author to specify the conditions under which their definition applies. In definition $(3),$ I've assumed that you are discussing $\mathbb R^2$ instead of, say, just quadrant $1.$
Equality $(4)$ can be derived from the definitions of the trigonometric functions, so it is mathematically true, so we call it a result/theorem.
In our previous discussion, I alluded to the fact that the trigonometric identity $$\sin^2(x)+\cos^2(x)\equiv1$$ is a theorem/result, not a definition.
An identity in $x$ is simply an equality that holds for all $x;$ in this case, the identity is true by derivation, in another case, it might be true by definition.
Since definition $(4)$ duplicates a known result, it leads to no inconsistency, and is merely redundant. On the other hand, $$\sin^2(x)+\cos^2(x):=7\tag5$$ is a bad definition because using it can lead to contradictory results.
You don't have to define $$\sin^2(x)+\cos^2(x)=1;$$ you merely have to quote (or perhaps derive/prove) it.