$1.($a$)$ An operator is said to be associative, if $($ AB$)$ C $=$ A $($BC$).$ for all A$.$ B and C in the domain under discussion. For instance, addition $(+)$ is associative over the integers, while subtraction $(-)$ is not. In propositional logic, both V and are associative operators. What can you say about the boolean operators $->$ and $+,$ as regards associativity? $($b$)$ The exclusive$-$or operator $()$ on two boolean variables A and B is defined as follows: A e B is true exactly when one of A and B is true. Draw the truth$-$table for the operator. Show that A B can be expressed using our usual boolean operators, viz., V$.$ A$,$ and'. $2.$ Let S $=\{+\text{'}).$ Argue that S is functionally complete for propositional logic. In other words, show that every well$-$formed formula in propositional logie can be formed using only the operators in S$.3.$ Is the following argument valid? Provide a proof of validity or an interpretation where it does not hold. $[($PV$($QAR$))($R$\text{'}$ V S$)$ A $($ST$\text{'})]->($TP$).$ You may not use the truth$-$table method in your analysis. $4.$ Show that the following arguments are tautologies, using rules of inference. $($a$)$ P$->($PAP$).($b$)($PVP$)$ P$.5.$ Consider the following verbal argument: It is not the case that if electric rates go up$,$ then go down. Nor is it true that either new power plants will be built or bills will not be late. Therefore, usage will not go down and bills will be late. Is it valid? usage will