Answer to Question #281784 in Discrete Mathematics for V.kathiravan

Question

show that ~Q,P—>Q=>~P in mathematical foundations of computer science

Answer 1

Let us construct a truth table for the compound proposition $(\sim Q\land (P\to Q))\to \sim P:$

$\begin{array}{||c|c||c|c|c|c|c||} \hline\hline P & Q & P\to Q & \sim Q & \sim P & \sim Q\land (P\to Q) & (\sim Q\land (P\to Q))\to \sim P \\ \hline\hline 0 & 0 & 1 & 1 & 1 & 1 & 1\\ \hline 0 & 1 & 1 & 0 & 1 & 0 & 1 \\ \hline 1 & 0 & 0 & 1 & 0 & 0 & 1\\ \hline 1 & 1 & 1 & 0 & 0 & 0 & 1\\ \hline\hline \end{array}$

It follows that the formula $(\sim Q\land (P\to Q))\to \sim P$ is a tautology, and hence $\sim Q, P\to Q\vdash\ \sim P.$