The user activity for finding informative meaningful objects needed to solve the problem is considered. To formalize this activity, set theory is used. The mathematical models of formulation of request, analysis of the issued documents and selection of information objects for the solution to the user problem are presented. In addition, mathematical models of intellectual activitiesof user and cognitive processes are developed. The experimental results of the search engines such as Google, Yandex, META, Rambler, Yahoo are given.