When reading mathematics how do you intuitively interpret the given objects such as sets, functions, sequences, etc? do you automatically visualize them graphically?
When dealing with sets $A, B, X, Y$ do you think of them represented as venn diagrams?
And whenever you come across functions $f:X\rightarrow{Y}$ given by some formula for $f(x)$, and for sequences $a_n:\mathbb{N} \rightarrow \mathbb{R}$ given by some formula for $a_n$, do you interpret $f(x)$ and $a_n$ as the points along the vertical axis?
I am self-studying Real Analysis and would like to be able to represent the given information in a more beneficial way rather than just memorizing symbols.
Most notation is contextual. Here is a rough standard: $\require{amsmath}$ \begin{align} &i,j,k,l,m,n,p\ldots&\text{integers}\\ &p,q,r,s,t\ldots&\text{real numbers}\\ &\mathbb{C},\mathbb{R},\mathbb{Q},\mathbb{Z},\mathbb{N}\ldots&\text{sets of numbers}\\ &X,Y,W,Z,\Omega,\Gamma,\Lambda\ldots&\text{main sets---e.g., linear spaces}\\ &A,B,C,U,V,S,T\ldots&\text{subsets of main sets}\\ &a,b,x,y,z,\alpha,\beta,\lambda,\mu,\omega\ldots&\text{elements of sets}\\ &\mathcal{A},\mathcal{B},\mathcal{C}\ldots&\text{sets of sets---e.g., filters, topologies}\\ &f,g,h,p,q,\alpha,\beta,\lambda,\mu,\pi\ldots&\text{functions}\\ &\Gamma,\Delta,\Phi,\Psi\ldots&\text{sets of functions} \end{align}
As for your question, think about how you would go about teaching the very thing you are reading. Then you will see the utility of notation. Furthermore, you will have a better understanding of why the author of the written material chose to write things in a certain way (maybe not the best way, in your opinion). It is up to you to rewrite things you would like to memorize in your own language with the caveat that likely others will dislike your own language and that some translation may be required when speaking to others.
I want to conclude that it is very important that mathematicians be allowed to have their own unique perspective and visualization. It is fine to ask how others visualize, but feel free to attempt to create your own visualization and run with it.