Linear and affine sets and relations have been formerly investigated by a great number of authors. The aim of the present paper is to put the subject into a new perspective by slightly improving and supplementing some of the earlier definitions and results. In the family \(\mathscr {P}(X)\) of all subsets of a vector space X over a field K, we shall consider the usual extensions of the linear operations to sets. Thus, \(\mathscr {P}(X)\) , with the ordinary set inclusion, forms a complete, partially ordered generalized vector space with null, zero and infinity elements \(\emptyset \) , \(\{0\}\) and X. Now, a subset A of the vector space X may be naturally called Moreover, a relation R on X to another vector space Y  over K (i.e., a subset R of the product set \(X\!\times \!Y\) ) may be naturally called linear (affine) if it is a linear (affine) subset of the product vector space \(X\!\times \!Y\) . These properties can also be expressed in terms of the values \(R\,(x)\) with \(x\in X\) . Some of the results obtained will be extended to super and hyper relations on X to Y  in a subsequent paper. However, for this, instead of \(\mathscr {P}\,(X\!\times \!Y)\) the more difficult power sets \(\mathscr {P}\,\bigl (\mathscr {P}\,(X)\times Y\,\bigr )\) and \(\mathscr {P}\,\bigl (\mathscr {P}\,(X)\times \mathscr {P}\,(Y)\,\bigr )\) have to be equipped with appropriate inequalities and linear operations.