# How to interpret objects and symbols in written mathematics?

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.

• There is no single answer, it depends on context. Some problems are better addressed graphically, and for others this is of no help or is just impossible. – Yves Daoust Feb 1 at 15:29
• Venn diagrams are handy for three sets but not more. Karnaugh tables remain practical up to six sets. – Yves Daoust Feb 1 at 15:34

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}