1. Construct a truth table for each of the given statements. Is the statement a tautology?...

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1. Construct a truth table for each of the given statements. Is the statement a tautology? Explain! (a) (P = Q) A (Q = R) = (P = R) (b) ( ΡAQ R)- (P- (Q^ 2. Verify each law of logic (by constructing the appropriate truth table) and then write each law in a tautology form. (a) DeMorgan's Law : - (PAQ) = PV ~Q (b) Rule for Direct Proof : (PAR= Q) = (R (P= Q)) (c) Addition Law: (P PV Q) = t where t is a tautology. 3. The following statement is not a tautology. Find a counterexample. (a) (PAQ= R) + (P= R) A (Q = R) (b) PVQ = P (c) P PAQ 4. For each of the following statements, check whether or not it is logically equivalent to the negation of P = Q. In each case, verify your answer. (a) - P = Q (b) - PAQ (c) PV ~Q 5. Reformulate each of the following so that it is logically equivalent to a statement that has only NOT and OR as connectives. (a) The converse of P= QVR (b) The contrapositive of PAQ= R. 1. Construct a truth table for each of the given statements. Is the statement a tautology? Explain! (a) (P = Q) A (Q = R) = (P = R) (b) ( ΡAQ R)- (P- (Q^ 2. Verify each law of logic (by constructing the appropriate truth table) and then write each law in a tautology form. (a) DeMorgan's Law : - (PAQ) = PV ~Q (b) Rule for Direct Proof : (PAR= Q) = (R (P= Q)) (c) Addition Law: (P PV Q) = t where t is a tautology. 3. The following statement is not a tautology. Find a counterexample. (a) (PAQ= R) + (P= R) A (Q = R) (b) PVQ = P (c) P PAQ 4. For each of the following statements, check whether or not it is logically equivalent to the negation of P = Q. In each case, verify your answer. (a) - P = Q (b) - PAQ (c) PV ~Q 5. Reformulate each of the following so that it is logically equivalent to a statement that has only NOT and OR as connectives. (a) The converse of P= QVR (b) The contrapositive of PAQ= R.