Drawing Arrow Diagrams & Explaining Domain, Codomain & Range

[lgmath]

Hi Folks,

Have a task , but no idea how to resolve.
Will appreciate any help.

Consider the function f: {5,6,7} →{A,B,C} given by f(5)=A, f(6)=C, and f(7)=B
Draw its arrow diagram and state its domain, codomain and range.
Is it (i) onto and (ii) one-to-one ? Explain. Does it have an inverse f^-1 and if so
what is it ?

Thanks.

 

[Evgeny.Makarov]

Hi, and welcome to the forum.

I think that your difficulty comes from not knowing the definitions of the concepts involved: domain. codomain and range, one-to-one and onto functions and the inverse function. I believe that once a person knows these definitions, applying them to the given function is straightforward. While it is possible that some helper would write these definitions with explanations and examples for you, all that he/she be doing is duplicating information from your textbook. In fact, your textbook must have been written by professionals and undergone an editing process and chosen by your professors, while helpers on this forum are volunteers. So my advice to you is to go back to your textbook and review the concepts from your problem. Read about any examples of these concepts. If, after doing this, you have a problem understanding any material or if you don't know how to apply it here, then please describe your exact difficulty.

Having a forum is not a substitute for reading a textbook. As Euclid said, "There is no Royal Road to geometry".

 

[lgmath]

Wow, nine lines of text and nothing I can take from it.
What a waste of space and time.
Thanks for attention anyways :)

 

[MarkFL]

You see, the difficulty arises for us when someone posts a question and does not show what their thoughts are on how to begin or what they have tried and where they are stuck.

We are then left not knowing how best to help. Do we post information that is readily available in a textbook, at the risk of wasting your time and ours?

We expect the heavy lifting to be done by the person asking the question, where we offer guidance. You will gain much more by taking the effort and time to show what you think you should do and what you have tried and then we can tell you why what you are doing is not working or offer a hint on how to proceed.

We would be doing little service by working the problem for you. :D

 

[CellCoree]

Hello,

Drawing arrow diagrams is a great way to visually represent functions and their relationships. In this case, we have a function f that maps elements from the set {5,6,7} to the set {A,B,C}. Let's start by drawing an arrow from each element in the domain to its corresponding element in the codomain. We would have an arrow from 5 to A, 6 to C, and 7 to B. This shows us the mapping of the function f.

Now, let's define the domain, codomain, and range of this function. The domain is the set of input values, in this case {5,6,7}. The codomain is the set of possible output values, which is {A,B,C} in this case. The range is the set of actual output values, which is {A,B,C} as well. This means that every element in the codomain has at least one corresponding element in the domain, making this function onto.

To determine if the function is one-to-one, we need to check if each element in the domain is mapped to a unique element in the codomain. In this case, we can see that 5 and 7 are both mapped to different elements in the codomain, but 6 is mapped to the same element as 7. Therefore, this function is not one-to-one.

As for the inverse function, f-1, we can only have an inverse if the function is both onto and one-to-one. Since this function is not one-to-one, it does not have an inverse function.

I hope this helps to clarify the concepts of domain, codomain, range, and the properties of a function. Good luck with your task!