 Hi and welcome to this session. I'm Kanika and I'm going to help you to solve the following question the question says Examine the validity of the following argument. S1 is P and negation of Q implies R S2 is P or Q, S3 is Q implies P and S is R Let's now proceed on with our solution We will first construct a truth table for All hypothesis and Conclusion in this question three Hypothesis are given to us first one is P and negation of Q implies R second one is P or Q and Third one is Q implies P and the conclusion is R So let's make a truth table for all hypothesis and conclusion Let P has truth value T, T, T, T, F, F, F, F and let Q has truth value T, T, F, F, T, T, F, F and let R has truth value T, F, T, F, T, F, T, F When Q has truth value T the negation of Q has truth value F When Q has truth value F the negation of Q has truth value T Now we will find the truth value of P and negation of Q If P has truth value T and negation of Q has truth value F then truth value of P and negation of Q is Both P and negation of Q has truth value T then truth value of P and negation of Q is T If both P and negation of Q has truth value F then truth value of P and negation of Q is F If P has truth value F and negation of Q has truth value T then truth value of P and negation of Q is F. Let's now find the truth value of all the hypothesis. First hypothesis is P and negation of Q implies R. If truth value of P and negation of Q is F and if truth value of R is T then truth value of P and negation of Q implies R is T, then we have F implies F, truth value of F implies F is T, then we have T implies T, truth value of T is t is t, then we have t implies f, truth value of t implies f is f, then we have f implies t, truth value of f implies t is t, then we have f implies f, truth value of f implies f is t, then we have f implies t, truth value of f implies t is t, and at last we have f implies f, so truth value of f implies f is t. Now we will find the truth value of P or Q. If both P and Q has truth value T then P or Q has truth value T. If P has truth value T and Q has truth value F then P or Q has truth value T. If P has truth value F and Q has truth value T then P or Q has truth value t. If both p and q has truth value f then p or q has truth value f. Now we will find the truth value of q implies p. If truth value of q is t and truth value of p is t then truth value of q implies p is t. Then we have f implies t, truth value of f implies t is t. Then we have t implies f, truth value of t implies f is f and at last we have f implies f, truth value of f implies f is t. Now we will find the truth value of the conclusion. Now the conclusion is r. So truth value of r is t f, t f, t f, t f. Now we will find the critical rows. Critical rows are those rows in which all the hypothesis are true. In first row all the hypothesis are true, in second row also all the hypothesis are true and in third row also all the hypothesis are true. Now we will check whether the conclusion is true or not in each critical row. Conclusion is true in first and third critical row but the conclusion is falls in second critical row. So this means that the given argument is invalid. So truth value of third table consists of three critical rows. The conclusion is falls in second critical row. Hence the given argument invalid. This is our required answer. Bye and take care. Hope you have enjoyed this lesson.