Is there a math function to find an element in a vector?

Solution 1:

Usually this is written as "Let $i$ be such that $x_i = j$." But that's not very compact. There isn't any standard notation for this that I know of. You could use $\arg \max_i [x_i = j ]$ which uses Iverson brackets to make things compact.

Solution 2:

If you want to talk about the $3$ in the vector $x=(1,2,3)$ then most people will just denote this element $x_3$ to indicate the third element of the vector.

If you are interested in the function that maps $x\to x_3$ then that function is denoted $\pi_3(x)$ and is called "the projection function (onto the third coordinate)". This function is pretty important in topology.

If you are interested in the function that tells you what index $k$ is, there isn't really a common notation for that because it's not necessarily a function. Besides, even when talking about it as a relation, in almost all circumstances if you know that $x$ contains a $3$, you can also just know which indices are $3$ and which are not as a consequences of knowing what x is