 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that for any three statements P, Q, R construct a truth table for P and Q or R. Let us proceed with the solution. We have to construct a truth table for the statement having symbolic notation P and Q or R. To construct its table, we first make the truth table for P, Q, R taking all the possible truth values of P, Q and R. In column 1 we write the possible truth values of P, in column 2 we write the possible truth values of Q and in column 3 we write the possible truth values of R. So here there are eight such possibilities. The first possibility is all statements are true. So in the first row we write the truth value P for P, Q and R. The second case can be two statements are true and one is false. So we have three possibilities T, T, F, T, F, T and F, T, T. So the next three rows will be P is T, Q is T, R is F, P is T, Q is F, R is T and P is F, Q is T, R is T. The third possibility can be two statements are false and one is true. So again we have three possibilities T, F, F, F, T and F, T, F. So in the next three rows P is T, Q is F, R is F, P is F, Q is F, R is T and P is F, Q is T, R is F. The last possibility is that all statements are false. So in the last row P, Q and R all of them are F. So in the truth table of P and Q or R we will first find the truth value of P and Q. To write truth values of P and Q we will take into consideration truth values of P and Q columns only and we will leave the truth values of R. Now to write the truth values of P and Q we take into consideration the conjunction table. So here we know that if P is true, Q is true, P and Q will be true. If P is true, Q is false, P and Q will be false. If P is false, Q is true, P and Q will be false. And if P is false, Q is false, P and Q will also be false. Now we insert these values into the truth table of P, Q, R. We make a new column in which we write the values for P and Q. In the first row both P and Q have truth value T and we know if P is true, Q is true then P and Q is also true. So we write truth value of P and Q as true that is T. In second row again both P and Q are true so P and Q will also be true. In the third row P is T and Q is F and we know if one of them is true and the other is false then the value for P and Q is false. So here in the third and the fourth row the value for P and Q will be false. Similarly for the fifth and the seventh row the value of P and Q is false. In the sixth and the eighth row the value of P and Q both are false. And as we know if the truth value of both P and Q are false then the truth value of P and Q is also false. So here in the sixth and the eighth row the value of P and Q will be F. Now we will write the truth values of P and Q or R. For this we will consider two columns the column for R and the column for P and Q. We know the truth value for this junction which is as follows. If P is true, Q is true, P or Q is true. If P is true, Q is false, P or Q is true. If P is false, Q is true, P or Q is true and if both P and Q are false then P or Q is also false. So using this we will write the truth values of P and Q or R. In the first row R is true, P and Q is also true so P and Q or R is true. In the second row R is false, P and Q is true so P and Q or R is true. Similarly in the third row R is true, P and Q is false so P and Q or R is true. As we know from the disjunction table that if two statements are true then the disjunction is also true. And if either of the two statements are false then the disjunction is true. So according to this in the fourth and the sixth row one of the values of R or P and Q is true. So the value of P and Q or R is also true. So lastly from the disjunction table we know that if both the values of P and Q are false then P or Q is also false. This means that the value of P and Q or R in the fifth, seventh and eighth row will be false. So this is the final truth table for P and Q or R. With this we complete our session. Hope you enjoyed the session.