composite functions:why is the target same for both functions

[Osman123]

have been given this question for a school assignment

"let f : W →T and g : S →W be two functions. Prove, relying on a mathematical reasoning, that the target of (f ◦g ) will be the target of f"

i am not sure how to explain or prove it can someone give me a little explanation i dont really understand this

 

[Dr.Peterson]

have been given this question for a school assignment

"let f : W →T and g : S →W be two functions. Prove, relying on a mathematical reasoning, that the target of (f ◦g ) will be the target of f"

i am not sure how to explain or prove it can someone give me a little explanation i dont really understand this

How is "target" defined in your class? Please quote the definition.

It isn't a standard term in my experience, but I am guessing it is what I know of as the "codomain". If so, then stating the definition you were given should almost answer the question. (Or it may reveal that the answer is really an arbitrary choice, and can't be proved. It may be necessary to see the definition you were given for the composite function, too.)

 

[pka]

have been given this question for a school assignment
"let f : W →T and g : S →W be two functions. Prove, relying on a mathematical reasoning, that the target of (f ◦g ) will be the target of f"
i am not sure how to explain or prove it can someone give me a little explanation i don't really understand this

In usual function notation, [imath]f:W\to T[/imath] is read [imath]f[/imath] is a function mapping set [imath]W\text{ to set }T[/imath].
In the mathematical notation, [imath]f\subseteq(W\times T)[/imath] or [imath]f[/imath] is a subset of the cross product of [imath]W\text{ with }T[/imath] such that each [imath]x\in W[/imath] is the first term of some pair in [imath]f[/imath] and no two pairs of [imath]f[/imath] have the same term. I assume that the set of second terms of [imath]f[/imath] is its target..
We apply the same notation to [imath]g: S\to W [/imath].
Can you now say that target [imath]f\circ g[/imath] is the target of [imath]f~?[/imath]

[imath][/imath][imath][/imath][imath][/imath]

 

[blamocur]

have been given this question for a school assignment

"let f : W →T and g : S →W be two functions. Prove, relying on a mathematical reasoning, that the target of (f ◦g ) will be the target of f"

i am not sure how to explain or prove it can someone give me a little explanation i dont really understand this

Not sure this can be proved if [imath]g(S) \neq W[/imath], i.e, if [imath]g[/imath] is not surjective.

 

[pka]

Not sure this can be proved if [imath]g(S) \neq W[/imath], i.e, if [imath]g[/imath] is not surjective.

The term target does not always imply a surjection. That is why Prof. Peterson asked for the definition used in the course.
The set of second terms of a function is usually called its range but some authors use target.

 

[blamocur]

The term target does not always imply a surjection. That is why Prof. Peterson asked for the definition used in the course.
The set of second terms of a function is usually called its range but some authors use target.

True, but if [imath]g(S)\neq W[/imath] then in a general case [imath](f\circ g) (S) \neq f(W)[/imath]

 

[Dr.Peterson]

True, but if [imath]g(S)\neq W[/imath] then in a general case [imath](f\circ g) (S) \neq f(W)[/imath]

I think pka either is saying something slightly different from what I said, or he has made a couple typos that may be confusing things. This is almost correct from my perspective:

In usual function notation, [imath]f:W\to T[/imath] is read [imath]f[/imath] is a function mapping set [imath]W\text{ to set }T[/imath].
In the mathematical notation, [imath]f\subseteq(W\times T)[/imath] or [imath]f[/imath] is a subset of the cross product of [imath]W\text{ with }T[/imath] such that each [imath]x\in W[/imath] is the first term of some pair in [imath]f[/imath] and no two pairs of [imath]f[/imath] have the same term. I assume that the set of second terms of [imath]f[/imath] is its target..
We apply the same notation to [imath]g: S\to W [/imath].
Can you now say that target [imath]f\circ g[/imath] is the target of [imath]f~?[/imath]

[imath][/imath][imath][/imath][imath][/imath]

Assuming the OP reports the definition I am expecting, the "target", or "codomain" (a name I dislike), is the set of all potential second entries in ordered pairs, that is, all of T in the example. It's very hard to accurately describe! (The range is the set of all actual second entries, as I describe it.)

The term target does not always imply a surjection. That is why Prof. Peterson asked for the definition used in the course.
The set of second terms of a function is usually called its range but some authors use target.

Rather than "range" here, I would say "codomain". The range is the subset of the codomain which is actually reached by the mapping, namely [imath]g(S)\subseteq T[/imath].

The tricky part of this "target" idea is that it is defined only as part of the definition of a function, and can't be determined from anything else we know about the function. That's why I think a proof will depend very much on the particulars of the definition.

Here are references to the definition I am assuming for "target" (Wikipedia only calls it "codomain" or "destination"):

"range of function" vs "target of function"?

Page 14 of Fundamentals of Computer Graphics states that if we have a function like this: ...the set that comes before the arrow is called the domain of the function, and the set on the right-hand ...

math.stackexchange.com

Codomain - Wikipedia

en.wikipedia.org

codomain - Wiktionary, the free dictionary

en.wiktionary.org

Definition:Codomain (Set Theory)/Mapping - ProofWiki

proofwiki.org