Using the density functional method in the general gradient approximation, the electronic and elastic properties of the systems inherent in the TiC–TiB$_2$ composite are simulated using the ABINIT software package. The electronic spectra of titanium diboride, titanium carbide, and the constructed supercell of TiC–TiB$_2$ composite are analysed. As established, the main feature of the electronic structure of the studied systems is the location of the Fermi level in the local minimum of the energy dependence of the electronic states density, which is formed mainly by 3$d$-electronic states of titanium atoms. As shown, a slight diffusion of carbon and boron atoms in the Ti–TiB$_2$ composite does not lead to radical changes in the electronic structure of the composite. The calculated bulk modulus of the TiC–TiB$_2$ composite supercell practically does not differ from ones for TiB$_2$.