How does distributive law in propositional logic apply to two bracketed terms?

Question

An example of an expression the question is talking about would be: $$(p ∧ ¬q)∨(¬p∨q)$$

Answer 1

Think $\color{blue}{(¬p∨q)}$ as one term, Apply the right distributive law:$$(\color{red}a\land \color{orange}b)\lor \color{blue}c\equiv(\color{red}a\lor \color{blue}c)\land (\color{orange}b\lor \color{blue}c)$$ We have the following: \begin{align} & (\color{red}p ∧ \color{orange}{¬q})∨\color{blue}{(¬p∨q)}\\ \equiv&((\color{red}p∨\color{blue}{(¬p∨q)}) ∧ (\color{orange}{¬q}∨\color{blue}{(¬p∨q)})\\ \end{align}

Answer 2

Let P be the statement "He's well-read". Let Q be the statement "He's a good sportsman".

(p∧¬q)∨(¬p∨q) logically equals to : "He's either well-read and bad sportsman, or he's not well-read and good sportsman". You can simply put it P v Q, where v(or) would be exclusive, which means that both can't be true at the same time (although in real life one can be a good sportsman and be well-read, rarely as it is, but still, for the sake of the arguement we'll take it as true.) So you can write it as> He's either well-read or a good sportsman.