(1) The function f is defined by f:x 1 - 4x, x R. Show that f is surjective (onto Z)
(2) The function g is defined by g:x 3x - 4, x Z. Show that g is not surjective (not onto Z)
I need help in understanding the term surjective/onto . I am aware that it means when the range of a function is the same as its co-domain, but how do i apply this in solving the above questions? Furthermore what exactly is the difference between the range of a function and the co-domain of a function?
EDIT: Tooooo sloooow
Originally Posted by ibnashraf
When you define a function, you define it going from a domain to a codomain. The set of values the come out of the function is the range. For example,
Is a function going from the integers to the integers. Here both the domain and the codomain are the set of integerss.
Now suppose that our f was, f(x)=2x, again going from integers to integers. Then the range in this case is the set of even integers, which is different from the codomain, which is the entire set of integers.
Then f is a surjection if for every y in Y, there exists a corresponding x in X such that f(x)=y.
Now for the first question, we have X=R and Y=Z. So to see if f is a surjection, it must hold that for all z in Z there exists some x in R such that f(x) = z i.e 1-4x=z. Thus all we have to do is for any z in Z, define x = (1-z)/4, that way we can see that f(x)=z and x is in R, which implies f IS a surjection.
Hope this helps, you should be able to try number 2 now.
You'll find that this is a useful page. Here's a quick summary:
Originally Posted by ibnashraf
- What can go in to a function is called the domain
- What may possibly come out of a function is called the codomain
- What actually comes out of a function is called the range
A function is said to be surjective (or onto) if the range and the codomain are one and the same set. In other words, all the things that could come out of the function do come out.
In question 1, the codomain is = the set of integers. So we're only allowing integers to come out. So we're not interested in any input which produces a non-integer output.
The question is: can we find enough inputs in the domain to ensure that every integer can be produced as an output? If we can, then the range and the codomain are the same, and the function is surjective.
To answer this question, suppose that we want the integer (where is any old integer!) to be output. Can we find a suitable input that will generate as an output? If other words, can we find an , such that , whatever integer represents?
The answer is yes. Solving for , we get:
which is a real number for all integers .
Now let's repeat the question for #2, noting that this time the input values are restricted to integers, rather than the reals. So, for any , can we find an integer for which ?
Solving for , we get
and obviously there are lots of values of (in fact, infinitely many of them) that will make a non-integer. For example, .
So this time the function is not surjective.
Wow, mine is the third reply. You have some reading to do!