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- Thread starter mahmoud2011
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Thanks

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Stephen Tashi

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If f(x) is a function with domain A and co-domain B then the image of f is set {y: f(x) = y for some x in A} and it is a subset of B. The co-domain of a function (it seems to me) is a somewhat arbitrary specification given as part of the definition of a function. Someone might say: "Let f(x) be a function from the reals into the reals defined by f(x) = x^2" without being specific that the image of f is the non-negative reals.

So what shall it mean to say that two functions are "equal"? Must they have the same co-domain or must they also have the same image? And do you really find a lot of material that talks about the equality of functions? There's lots of material about how f(x) = g(x) for all x, which stops short of saying that f and g are "equal" functions.

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If f(x) is a function with domain A and co-domain B then the image of f is set {y: f(x) = y for some x in A} and it is a subset of B. The co-domain of a function (it seems to me) is a somewhat arbitrary specification given as part of the definition of a function. Someone might say: "Let f(x) be a function from the reals into the reals defined by f(x) = x^2" without being specific that the image of f is the non-negative reals.

So what shall it mean to say that two functions are "equal"? Must they have the same co-domain or must they also have the same image? And do you really find a lot of material that talks about the equality of functions? There's lots of material about how f(x) = g(x) for all x, which stops short of saying that f and g are "equal" functions.

That is what I know But why Some books write that the equality of co-domains is necessary .

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Stephen Tashi

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That is what I know But why Some books write that the equality of co-domains is necessary .

As I said, I am not familiar with any material where the equality of functions is an important issue. Can you give specific examples of books that want the co-domains to be equal?

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As I said, I am not familiar with any material where the equality of functions is an important issue. Can you give specific examples of books that want the co-domains to be equal?

proofs and Fundamentals : A first course in Abstract Mathematics , Ethan D.Bloch

- #6

Stephen Tashi

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proofs and Fundamentals : A first course in Abstract Mathematics , Ethan D.Bloch

I don't have that book. Does he do any proofs that involve showing two functions are "equal"? When he does them, how does he handle the part about the co-domains?

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Proof. (Argumentation)

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Therefore the domain of f is the same as the domain of g.

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(argumentation)

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Therefore the codomain of f is the same as the codomain of g.

Let a be in the domain of f and g.

..

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(argumentation)

...

Then f (a) = g(a).

Therefore f = g.

- #8

Stephen Tashi

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(argumentation)

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Therefore the codomain of f is the same as the codomain of g

Can he give any sort of argumentation about the codomains being the same except that the codomains were specified to be the same when the functions were defined?

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Can he give any sort of argumentation about the codomains being the same except that the codomains were specified to be the same when the functions were defined?

No , but in his analysis book he says that in principle changing co-domains changes the function . and I see that it is not necessarily the case , that is for example if B ≠ f(A) m doesn't mean that these two functions are not equal f : A → B and f : A → f(A) .

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- #10

Stephen Tashi

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doesn't mean that these two functions are not equal f : A → B and f : A → f(A) .

That statement has too many negations for me to understand!

The technicality here is what is meant by "equal" in some mathematical context. (For example, equality of sets A and B has a different definition that equality of real numbers A and B.) This is complicated by the fact that we use "equal", "the same" , "identical" etc. in ordinary speech. If Bloch wished, he could make a distinction between the definition of two functions being "identical" and two functions being "equal". When he talks about the technicality of the co-domains, does he use the word "equal"?

- #11

Deveno

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f:A→B and

f:A→f(A)

to be "the same function".

the problems arise when we consider:

gf:A→C, where f:A→B and g:B→C, we don't want a "type-mismatch" in composing g and f.

this happens in, say computer programming, where you declare the variable type of function arguments.

so if x is of type "integer", and f:x→ x*x, we could declare f(x) to be of type "natural number", but then g might not process x*x properly if its input variable type is "integer".

in other words, if you restrict the co-domain of f to f(A), gf might not be defined without considering the (clearly related) function g|

in practice, this rarely comes up, but defining compositions can get really complicated if you have to include the various extensions and restrictions of the domain and co-domain.

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