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

Question #281784

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


1
Expert's answer
2021-12-22T07:22:49-0500

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


PQPQQPQ(PQ)(Q(PQ))P0011111011010110010011110001\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 (Q(PQ))P(\sim Q\land (P\to Q))\to \sim P is a tautology, and hence Q,PQ P.\sim Q, P\to Q\vdash\ \sim P.

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