In this paper, we give a mathematical foundation of diagramming methods that is used for describing basic elements and their relationship. We construct a mathematical structure of class and object diagrams so that some mathematical operations on the diagrams can be naturally defined. The paper shows that a set of class diagrams and a set of object diagrams are both closed under these operations, and that interpretation mapping from a set of class diagrams to a corresponding set of object diagrams preserves the class diagrams operations such as union and intersection.