1. ## easy question about "into" functions

what makes a function an "into" function?

is it a function which is not "onto?"

2. Originally Posted by jmedsy
what makes a function an "into" function?

is it a function which is not "onto?"
More precisely, we say that a function f is from a set A "into" a set B, if the range of the function is a (proper) subset of B. If the range is equal to B, that's when we say the function is onto.

3. thanks, that clears it up

4. Originally Posted by jmedsy
what makes a function an "into" function?

is it a function which is not "onto?"

A function $\displaystyle f:A\rightarrow B$ is "into" (i.e., injective or 1-1) if for $\displaystyle a,a'\in A\,,\,\,f(a)=f(a')\,\Longrightarrow\,a=a'$ , and it is onto (i.e., suprajective) if for all $\displaystyle b\in B$ there exists $\displaystyle a\in A\,\,\,s.t.\,\,\,f(a)=b$.

Read carefully the above definitions, chew them slowly, draw yourself some diagrams and examples to try to fully understands them...and go on with your mathematical life.

Tonio

Ps. Remember! In mathematics, "to all" means TO ABSOLUTELY ALL, without any exception, and "There exists" means "there exists AT LEAST one (but perhaps some (many) others, too)"...

5. Originally Posted by tonio
A function $\displaystyle f:A\rightarrow B$ is "into" (i.e., injective or 1-1) if for $\displaystyle a,a'\in A\,,\,\,f(a)=f(a')\,\Longrightarrow\,a=a'$ , and it is onto (i.e., suprajective) if for all $\displaystyle b\in B$ there exists $\displaystyle a\in A\,\,\,s.t.\,\,\,f(a)=b$.

Read carefully the above definitions, chew them slowly, draw yourself some diagrams and examples to try to fully understands them...and go on with your mathematical life.

Tonio

Ps. Remember! In mathematics, "to all" means TO ABSOLUTELY ALL, without any exception, and "There exists" means "there exists AT LEAST one (but perhaps some (many) others, too)"...
I don't think "into" and "injective" are the same thing. "injective" and "1-1" mean the same thing. but "into" is something different

6. i'm under that impression, but I'd like to verify that before i turn in this assignment.

that is, one-to-one and into are not the same

7. Originally Posted by Jhevon
I don't think "into" and "injective" are the same thing. "injective" and "1-1" mean the same thing. but "into" is something different

Yes, maybe so....though it usually is a given that if the function is $\displaystyle f:A \rightarrow B$ then the range of f is "into" B.
Perhaps a language barrier here...thanx.

Tonio

8. Originally Posted by Jhevon
More precisely, we say that a function f is from a set A "into" a set B, if the range of the function is a (proper) subset of B. If the range is equal to B, that's when we say the function is onto.
I, along with tonio, is also under impression that "into" means "injective". I looked at Wikipedia, MathWorld and PlanetMath, but this is nowhere stated explicitly, even though "onto" is explicitly associated with surjection.

Nevertheless, Wikipedia and MathWorld, as well as most other places that I can remember call a function with a domain A and a codomain B, "a function from A to B" (notation: f: A -> B). I believe this is the standard terminology. Also, such arbitrary function is said to map elements of A into elements of B, but this is used for elements; I believe it is different from mapping A into B. Wikipedia also has the expression "a function on A into B"; I am not sure about that.

Finally, I don't think I've ever seen "into" mean that the range of the function is a proper subset of the codomain. For example, injection f: A -> B is sometimes called "inclusion", or "embedding of A into B". I am pretty sure that the same terminology would be used if f is surjection as well.

I would also add that inquiring into a definition, especially when it is genuinely unclear, is a legitimate thing. It is quite different from asking for a help with a problem, which may or may not be legitimate.

9. In my experience a mapping $\displaystyle f:X\mapsto Y$ is said to be "into" when it can be either onto or not onto. So a nondescript real function is said to map the reals "into" the reals. I'm not sure if this is universally true.

10. Originally Posted by Drexel28
In my experience a mapping $\displaystyle f:X\mapsto Y$ is said to be "into" when it can be either onto or not onto. So a nondescript real function is said to map the reals "into" the reals. I'm not sure if this is universally true.
i agree. it can be used for both. it's just that if the function is known to be onto, they use the word "onto", otherwise, they say "into". I'm pretty sure "into" does not mean "injective" though. i've never seen it used that way.