This is like the notes made when reading mathematics by someone who is trying to figure out the link between the mathematical wisdom and his own problems in mind. Tuples are ordered sets, sets the order of whose elements are essential, thus containing an additional dimension of information. The graph of a function is also a set, only with a correspondence in between its elements. A projection is a function assciating a point (graph) with its coordinates, while the association of a coordinate to a point (graph) containing it is also a function. These functions differ from the other ones in that the image is a product of sets one member of which is the inverse image. While the product operation can be interpretted as a one-to-one correpondence from a certain sets of elements to one set of tuples consisting of a complete ordering of the elements (thus unique), a projection is like a 'sub-correspondence' where is image is a subset of the inverse image of the product operation. A relationship or restriction can and should 'always be explicitly listed' as a correspondence. The image of a function or a correspondence is the image of a projection function from the graph of the function or correspondence to its component containing the image of the function or correspondence. An operation is also a correspondence, e.g. product operation, addition operation, projection operation, linear operation. An equation is in the language of set theory a correspondence, yet rather a special one. What is the definition of an equation in connection with a correspondence? This and the question of what a solution is should be of fundamental importance to the understanding of fixed-point theory which lies in the center of both optimization and equilibrium analysis. If an equation is simply a correspondence, what is a solution? an inverse-image? Maybe to say that a solution to a particular equation exists is to say that an element is in the image of that particular correspondence. In this sense, existence of something means the something's being inside a particular set, a 'membership' status. The 'bounded' linear operation in convexity is also a correspondence. It became clear from the comment
that the 'membership' status, i.e. being inside a particular image set, is a relation that something is a solution of a particular correspondence / equation. A relation is a different concept from a correspondence. The presence of an element inside a set is a relation. In a sense a correspondence is also a relation (maybe not binary?). So is an equation.