Let \(P = \{q \in \mathbb{N}|1 < q < 4\}\), \(Q = \{q \in \mathbb{W}|0 \leq q < 2\}\) and \(R = \{q \in \mathbb{N}|q < 3\}\). Then show that \(P \times (Q \cap R) = (P \times Q) \cap (P \times R)\)
Answer:
To prove:
\(P \times (Q \cap R) = (P \times Q) \cap (P \times R)\)
Explanation:
\(Q \cap R = \{\)\(\}\)
\(P \times (Q \cap R) = \{\)\(\}\)
\(P \times Q = \{\)\(\}\)
\(P \times R = \{\)\(\}\)
\((P \times Q) \cap (P \times R) =\{\)\(\}\)
As a result, \(P \times (Q \cap R) = (P \times Q) \cap (P \times R)\)
Hence, we proved.