We now need to justify the correctness of the design against the C-element specification in Fig. 8a. Given a starting configuration of input X, Y both 1, and any starting output, the C-element should eventually output 1. Similarly, if X, Y are both 0, on any initial output our final output should be 0. For any other configuration of X, Y, the output signal should remain the same. Thus, given an input \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{hi},Y_{hi}$$\end{document}Xhi,Yhi, and crucially any starting output configuration Z, we wish to reach a state where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{hi}$$\end{document}Zhi has at least M molecules where M is a population threshold, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo has 0 molecules. Similarly, for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{lo}, Y_{lo}$$\end{document}Xlo,Ylo and any Z we should to see a presence of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_{lo}$$\end{document}Zlo after some time t. For all other configurations of the inputs we wish the output species to remain the same.