MHB Understanding Mapping: Domain, Range & W Subset of C

[nacho-man]

Hi,
This is a little different from most questions in that it's not something I want to solve, but rather hoping I could get a little clearer explanation on
what mapping is and why/when we use it?

I'm soon to go over conformal maps, but I don't think i understand anything to do with mapping as of yet. If someone could break it down in english, backed with some mathematic intuition that would be so very helpful, because mathematical jargon just confuses me, and that's all textbooks seem to use.

I have added two images.
The one with the domain and range, I'm assuming is just saying that a specific value in the domain corresponds to a specific value in the range. Is this the essence of mapping?as for the other image, What is being referred to by W when it says W $\subset C$
I know C is referring to the complex plane. I am not sure what this information is telling me. could someone give me an example of a function where W is a subset of the complex plane, and when W isn't? (After having defined what W is )

Thank you! you're always so much help mhb

 

[Sudharaka]

Hi,
This is a little different from most questions in that it's not something I want to solve, but rather hoping I could get a little clearer explanation on
what mapping is and why/when we use it?

I'm soon to go over conformal maps, but I don't think i understand anything to do with mapping as of yet. If someone could break it down in english, backed with some mathematic intuition that would be so very helpful, because mathematical jargon just confuses me, and that's all textbooks seem to use.

I have added two images.
The one with the domain and range, I'm assuming is just saying that a specific value in the domain corresponds to a specific value in the range. Is this the essence of mapping?as for the other image, What is being referred to by W when it says W $\subset C$
I know C is referring to the complex plane. I am not sure what this information is telling me. could someone give me an example of a function where W is a subset of the complex plane, and when W isn't? (After having defined what W is )

Thank you! you're always so much help mhb

I visualize a function (a mapping) as a rule between two sets. For each element in the first set (which we call the domain) there is a unique element in the second set (which we call the range). A detailed by easy to follow discussion about functions can be found >>here<<.

Now \(W\subset\mathbb{C}\) tells you that \(W\) is a subset of \(\mathbb{C}\). It just means that each element of the set \(W\) lies in set \(\mathbb{C}\). An example would be the set of real numbers; \(W=\Re\). This is a subset of the complex numbers.

 

[alyafey22]

A mapping takes a value from some set $D$ and gives the image in another set $R$. In the complex plane mapping is interesting , take for example the set $|z|\leq 1$ . Now take for example the following function $f(z)=z$ . So we are mapping $z$ into itself. So if $|z|\leq 1$ is a disk with radius $1$ the mapping will be also a disk with radius $1$. Now take for example the function $f(z)=2z$ . The mapping will differ because we have $2$ so if we let for example $z=1$ we have $f(1)=2$ .So here we have $D:= |z|\leq 1$ , $\, R:= |z|\leq 2$.