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I am reading about when to function are equal or identical. As per the source through which i am studying two function f and g are equal and identical if they satisfy following conditions.

1) Domain of f should be equal to Domain of g.

2) Range of f should be equal to Range of g.

3) f(x)=g(x), for every x belonging to their common domain.

I am little confused with third condition, let D denote the Domain of function f , as per the condition 1 if g is equal to f then the domain of g is D itself.

So common domain will be D. So instead of writing Domain of f or Domain g why they have used the word common domain or I am missing some crucial point?

My another question is, are these three condition are necessary but not sufficient for stating to function are equal?

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"So instead of writing Domain of f or Domain g why they have used the word common domain?"

You have a point here in the sense that the third condition is given as some enlargement of the first condition. But that does not really harm, does it?

Let me give you a more handsome definition of the statement $f=g$ where $f$ and $g$ are both functions.

$f$ and $g$ have the same domain and for every $x$ that is an element of that domain we have $f(x)=g(x)$.

This definition is equivalent with the one in your question. Actually I put the conditions 1) and 3) together and leave condition 2) out. This because condition 2) is automatically satisfied if the conditions 1) and 3) are satisfied, hence is redundant. This definition is practicized in e.g. set theory.

Another point is that definitions are usually given as "if" statements, but should be read as "if and only if" statements (so necessity and sufficiency).


P.S. In certain areas of mathematics (e.g. categories) more is demanded for functions $f$ and $g$ to be equal. The condition that I gave is then accompanied with the condition that $f$ and $g$ have a common codomain (which is - at least in this context - not the same thing as range). Here by statement "$f$ and $g$ have a common codomain" is meant that the codomain of $f$ is the same as the codomain of $g$. So it is not the (much weaker) statement that the codomains of $f$ and $g$ have a non-empty intersection.

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  • $\begingroup$ I understand the meaning of codomain and range but still I am not able to understand the last statement you made i.e "The condition that I gave is then accompanied with the condition that f and g have a common codomain (which is - at least in this context - not the same thing as range).". Can you please elaborate further? $\endgroup$ – Thinker Apr 6 at 11:22
  • $\begingroup$ By saying that "$f$ and $g$ have a common codomain" I mean that the codomain of $f$ is exactly the same as the codomain of $g$. So if $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb Q$ then it is immediate that according to the second definition we have $f\neq g$. However, it can still happen that $f(x)=g(x)$ for every $x\in\mathbb R$ and in that case we have $f=g$ according to the first definition. $\endgroup$ – drhab Apr 6 at 11:39
  • $\begingroup$ Ok, so you mean in some case of mathematic for two functions to be equal both should have same codomain as well as range. $\endgroup$ – Thinker Apr 6 at 16:15
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    $\begingroup$ The range must always be the same, and this will be (automatically) be the case if the condition mentioned in my answer is satisfied. On some areas of mathematics that is enough (e.g. set-theory. But there areas where that is not enough. Also it is demanded then that the codomains are the same (e.g. categories). $\endgroup$ – drhab Apr 6 at 17:12
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Remember that, for $f=g$ to hold, the domains of $f$ and $g$ must be the same. Then all that the phrasing "common domain" really means is the domain of the pair of functions.

Another way of phrasing it: functions $f : S \to T, g : S' \to T'$ are equal if and only if

  • $S = S'$
  • $T = T'$
  • $f(x)=g(x)$ for all $x$ (in $S$ or $S'$ - it doesn't matter which since $S=S'$)

As for your question about necessary/sufficient - this is the definition of function equality. If there is an implication to be involved, then, it's the "if and only if" type, i.e. this is necessary and sufficient. Granted, what the definition of equality means might vary depending on the context (for example in my Fourier analysis class we altered the third premise to be that "$f(x)=g(x)$ for all but finitely many $x$").

In short, this is not a list of conditions for equality. This is equality, the definition of it (up to the relevant context).

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  • $\begingroup$ Understood the answer for my first question but I still have one query left ,"So for what context this equality conditions for functions holds? " $\endgroup$ – Thinker Apr 6 at 8:32
  • $\begingroup$ It's held in pretty much every context I've seen it in - said Fourier analysis class made a point of stating it explicitly to avoid confusion, for the reason I suppose it's one of the rare exceptions, so I would assume it holds unless explicitly told otherwise. $\endgroup$ – Eevee Trainer Apr 6 at 8:34

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