Is it necessary to have a formula to define a function?

[san203]

i realize there is a similar thread here But the questions are not the same.
1.)Function is a relation but i don't know what relation exactly means. Its supposed to be a condition associating two objects but also takes in the quantitative factor in maths?

2.)Anyways, functions can be defined as equations but not all of them are expressed as equations.
Can someone give me an example and state why?

3.)When one associates an element with another element, it doesn't necessarily imply equality but functions are defined as F(x) = y, where x and y are the respective elements . Doesn't this become an equation even though x and y itself are not similar things.

##$But then again when the elements are numbers, i see that the function(condition) f itself becomes equal to y e.g. :- let f be the condition where every value of x from R subtracted by 2. then y= x-2. I thought y and x were related by f but here y = f(x)$##

 

[SteamKing]

Maybe this article will help clear up your confusion:

http://www.purplemath.com/modules/fcns.htm

 

[san203]

I have actually gone through those articles. I have read a few books and top 10 results on google but those repeat the same definition. "Function is a device. Insert Input get output."
My questions is a bit different.

 

[adjacent]

I have actually gone through those articles. I have read a few books and top 10 results on google but those repeat the same definition. "Function is a device. Insert Input get output."
My questions is a bit different.

For example,
$f(x)=2x+1$
2x+1 is a function of x
Where is input?
$f(input)=2(input)+1-->Output$
That's how a function normally works.

 

[HallsofIvy]

i realize there is a similar thread here But the questions are not the same.



1.)Function is a relation but i don't know what relation exactly means. Its supposed to be a condition associating two objects but also takes in the quantitative factor in maths?

I'm not sure what you mean by "quanitative factor" here. A relation, between sets P and Q, is a subset of PXQ. That is, it is a collection of ordered pairs, in which the first member of each pair is a member of set P and the second member of each pair is a member of set Q. There is NO reqirement that P and Q be sets of numbers.

2.)Anyways, functions can be defined as equations but not all of them are expressed as equations.

No, that is not true. (And I don't how you distinguish "defined as equations" from "expressed as equations".) A function, from set P to set Q, is a relation between P and Q such that no two different pairs have the same first member. For example \{(x, x^2)\}, for x an integer, is a function but the relation \{x^2, x)\} is not: (4, 2) and (4, -2) are two pairs in that relation with the same first member. Again, there is NO requirement that P and Q be sets of numbers.

Notice that, given a relation, {(x, y)}, reversing the pairs, {(y, x)}, is again a relation. If {(x, y)} is function, {(y, x)} is not necessarily a function. That is why we say a relation is "between" two sets while a function is "from" one "to" the other.

Can someone give me an example and state why?

3.)When one associates an element with another element, it doesn't necessarily imply equality but functions are defined as F(x) = y, where x and y are the respective elements . Doesn't this become an equation even though x and y itself are not similar things.

I guess you could call that an "equation" in very extended way. However, there is NOT necessairily a "formula" expressing "F". For example, I can define a function from the alphabet to itself by {(a, b), (c, b), (b, q)}. I could then define F(a)= b, F(b)= q, F(c)= b. But there is no "formula" in the sense in which you seem to be thinking.

##$But then again when the elements are numbers, i see that the function(condition) f itself becomes equal to y e.g. :- let f be the condition where every value of x from R subtracted by 2. then y= x-2. I thought y and x were related by f but here y = f(x)$##

In any relation, we can write "y= f(x)" as above but the "f" is then given from the set of pairs. Even "when the elements are numbers", I can define a function pretty much at random:
{(1, 3), (-2, 5), (7, -100)}. Now I can write 3= f(1), 5= f(-2), and -100= f(7). But I am not going to be able to find a simple "formula", in the sense of arithmetic operations, for that function.

What is true is that when we are dealing with infinite sets, we cannot just write out the pairs in the function. In order to deal with them or even talk about them, we have to have some more general way of specifying the pairs- such as writing out a "formula".

But that is a limitation on us, not on "functions". There exist functions so complicated we cannot write "formulas" for them but then we really cannot work with those functions so we simply ignore them.

(And there are "intermediate" functions such as the "Bessel function" which is defined as the solution to "Bessel's differential equation". We cannot write this function as a formula in terms of simpler operations. We (approximately) determine its values by doing a numerical solution to the differential equation.)

 

[san203]

I'm not sure what you mean by "quanitative factor" here. A relation, between sets P and Q, is a subset of PXQ. There is NO requirement that P and Q be sets of numbers.

Sorry if i didn't express myself clearly but by those "quantitative Factors" i meant not the type of relation , i.e. b/w numbers or letters or anything but the number of such ordered pair connected through that relation.

(And I don't how you distinguish "defined as equations" from "expressed as equations".)

I didnt mean to. Sorry again.

A function, from set P to set Q, is a relation between P and Q such that no two different pairs have the same first member. For example \{(x, x^2)\}, for x an integer, is a function but the relation \{x^2, x)\} is not: (4, 2) and (4, -2) are two pairs in that relation with the same first member. Again, there is NO requirement that P and Q be sets of numbers.

Notice that, given a relation, {(x, y)}, reversing the pairs, {(y, x)}, is again a relation. If {(x, y)} is function, {(y, x)} is not necessarily a function. That is why we say a relation is "between" two sets while a function is "from" one "to" the other.

Although i completely understand what is being told here, this isn't the answer to my original question.

I guess you could call that an "equation" in very extended way.
However, there is NOT necessarily a "formula" expressing "F". For example, I can define a function from the alphabet to itself by {(a, b), (c, b), (b, q)}. I could then define F(a)= b, F(b)= q, F(c)= b. But there is no "formula" in the sense in which you seem to be thinking.

In any relation, we can write "y= f(x)" as above but the "f" is then given from the set of pairs. Even "when the elements are numbers", I can define a function pretty much at random:
{(1, 3), (-2, 5), (7, -100)}. Now I can write 3= f(1), 5= f(-2), and -100= f(7). But I am not going to be able to find a simple "formula", in the sense of arithmetic operations, for that function.

What is true is that when we are dealing with infinite sets, we cannot just write out the pairs in the function. In order to deal with them or even talk about them, we have to have some more general way of specifying the pairs- such as writing out a "formula".

But that is a limitation on us, not on "functions". There exist functions so complicated we cannot write "formulas" for them but then we really cannot work with those functions so we simply ignore them.

Thank you. Helped clarify a lot of thoughts. This means that we can define any functions without the need write Formulas. And the elements just have to be related somehow under that particular relation(function).