gate CSE discrete mathematics

Question

we have to check whether the statement is equivalent or not. For that we try to find a true false case and if it exists then that will imply that it is not equivalent. Now for creating True in LHS we say that there exists an x such that either P(x) or Q(x) is true for it. for RHS can we say that there exists an x such that p(x) is false and there exists another x for which Q(x) is false.??????? Since we know that both the x in RHS are not similar and there is no condition in LHS for both P(x) and Q(x) to be true simultaneously and any one of them being true at a time can do the job. Thanks

Answer 1

This statement is the distributive property which states that "There exists" quantifier is distributed over OR operations and it is fully distributive,hence this bi implication is true.

x in lhs and rhs is similar. x is bounded to some domain. Say x represents student in a class.If you want them to be different then you need to use different variables like x and y. As there are no different variables so given that your LHS is true ,you cant make RHS false or vice versa