When two functions are equal ?

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In summary: Your language isn't precise enough to capture what Bloch is trying to tell you at this point.In summary, some books state that when proving the equality of two functions, it is necessary to show that their co-domains are also equal. However, this may not always be the case and is dependent on the specific context and definition of "equal" being used. In some cases, it may be enough to show that the domains and images of the functions are equal. The issue of co-domains becomes more important when considering compositions of functions.
  • #1
mahmoud2011
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in some books when we have a function f : A → B , and g : C → D , to prove that they are equal it proves that the domains are equal and the codomans are equal , in other words A=C and B=D , and that f(a) = g(a) , for all a element of A , I see that proving that codomains are equal is not necessarily important that is because f and g are sets of ordered pairs , that are subsets of A x B and C x D respectively , and I proved that they are equal if and only if A = C , and f(A) = f(C) , in other words A = C and f(a) = g(a) for all a element of A , is that right ?

Thanks
 
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  • #2
The "co-domain" of a function isn't the same as the "image" of the function.
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.
 
  • #3
Stephen Tashi said:
The "co-domain" of a function isn't the same as the "image" of the function.
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 .
 
  • #4
mahmoud2011 said:
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?
 
  • #5
Stephen Tashi said:
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
mahmoud2011 said:
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?
 
  • #7
he writes when giving a proof that two functions are equal we write the proof in that form

Proof. (Argumentation)
.
.
.

Therefore the domain of f is the same as the domain of g.
.
.
.
(argumentation)
.
.
.
Therefore the codomain of f is the same as the codomain of g.

Let a be in the domain of f and g.
..
.
.
(argumentation)
...
Then f (a) = g(a).
Therefore f = g.
 
  • #8
mahmoud2011 said:
(argumentation)
.
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?
 
  • #9
Stephen Tashi said:
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
mahmoud2011 said:
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
if all we were ever concerned about was just single functions, that there would be little harm in declaring:

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|f(A), that is, g restricted to f(A).

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.
 

What does it mean when two functions are equal?

When two functions are equal, it means that they have the same output for every input value. This means that if you were to graph both functions on a coordinate plane, they would produce the same exact line.

How can you tell if two functions are equal algebraically?

To determine if two functions are equal algebraically, you can set the two functions equal to each other and solve for the variable. If the solution is the same for both functions, then they are equal.

Can two functions be equal but have different names?

Yes, two functions can have different names but still be equal. Just like how two people can have different names but still have the same height.

Are there any limitations to when two functions can be considered equal?

Yes, there are limitations to when two functions can be considered equal. For example, if one function has a domain restriction that the other function does not have, then they cannot be considered equal. Additionally, two functions cannot be considered equal if they have different degrees or exponents.

What is the importance of understanding when two functions are equal in the field of science?

Understanding when two functions are equal is important in science because it allows us to make accurate predictions and analyze data. By knowing that two functions are equal, we can confidently use one function to represent the other and simplify complex equations. This can also help in identifying patterns and relationships between different variables.

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