$\begin{align}  & \text{(i)Let }P,Q\text{ and }R\text{ be set, prove that } \\ & P\cap (Q\cup R)=(P\cap Q)\cup (P\cap R) \\ & \text{From the L}\text{.H}\text{.S} \\ & \text{Let }x\in P\cap (Q\cup R) \\ & x\in P\text{ and }x\in (Q\cup R) \\ & x\in P\text{ and }x\in Q\text{ or }x\in R \\ & x\in P\text{ and }x\in Q\text{ or }x\in P\text{ and }x\in R \\ & x\in (P\cap Q)\text{ or }x\in (P\cap R) \\ & x\in (P\cap Q)\cup (P\cap R) \\ & \therefore (P\cap Q)\cup (P\cap R)\subseteq P\cap (Q\cup R)---(i) \\ & \text{From the R}\text{.H}\text{.S} \\ & \text{Let }y\in (P\cap Q)\cup (P\cap R) \\ & y\in (P\cap Q)\text{ or }(P\cap R) \\ & y\in P\text{ and }y\in Q\text{ or }y\in P\text{ and }y\in R \\ & y\in P\text{ and }y\in Q\text{ or }y\in R \\ & y\in P\text{ and }y\in (Q\cup R) \\ & y\in P\cap (Q\cup R) \\ & P\cap (Q\cup R)\subseteq P\cap (Q\cup R)----(ii) \\ & \therefore P\cap (Q\cup R)=P\cap (Q\cup R) \\ &  \\ & (ii)\text{Let }P,Q\text{ and }R\text{ be set, prove that } \\ & P\cup (Q\cap R)=(P\cup Q)\cap (P\cup R) \\ & \text{let }x\in P\cup (Q\cap R) \\ & x\in P\text{ or }x\in (Q\cap R) \\ & x\in P\text{ or }x\in Q\text{ and }x\in R \\ & x\in P\text{ or }x\in Q\text{ and }x\in P\text{ or }x\in R \\ & x\in (P\cup Q)\text{ and }x\in (P\cup R) \\ & x\in (P\cup Q)\cap (P\cup R) \\ & (P\cup Q)\cap (P\cup R)\subseteq P\cup (Q\cap R)---(i) \\ & \text{From the R}\text{.H}\text{.S} \\ & \text{Let }y\in (P\cup Q)\cap (P\cup R) \\ & y\in (P\cup Q)\text{ and }(P\cup R) \\ & y\in P\text{ or }y\in Q\text{ and }y\in P\text{ or }y\in R \\ & y\in P\text{ and }y\in Q\text{ or }y\in R \\ & y\in P\text{ and }y\in (Q\cup R) \\ & y\in P\cap (Q\cup R) \\ & P\cap (Q\cup R)\subseteq (P\cup Q)\cap (P\cup R)---(ii) \\ & \therefore P\cap (Q\cup R)\subseteq (P\cup Q)\cap (P\cup R) \\\end{align}$