Is x Px Qx equivalent to x Px x Qx Justify your answerSoluti

Is x (P_(x) Q_(x)) equivalent to x P(x) x Q(x)? Justify your answer.

Solution

Let\'s consider M to be a structure in our math language which universe is {0,2}.

Let P(x)={0} and Q(x)={2}.

Now you can show that not all of the statement is true in M. Relations with 1 variable are subsets of this universe. In this inference you can look which elements in M satisfy P(x) and Q(x).

Another example could be to replace 0 by i and 2 by j where for all P(x) = {i} and Q(x) = {j} in the {i, j} universe.