Question

If \(A, B\) and \(C\) are sets, then \(A \times(B \cap C)=(A \times B) \cap(A \times C)\)

Short answer

If \(A\), \(B\) and \(C\) are sets, then it is indeed true that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\). This has been proved by showing that each side of the equation is a subset of the other.

Step-by-step solution

Understanding The Terms

In set theory, the Cartesian product of \(A\) and \(B\), denoted by \(A \times B\), is the set of all ordered pairs \((a, b)\) where \(a\) is in \(A\) and \(b\) is in \(B\). The intersection of \(B\) and \(C\) is denoted by \(B \cap C\) and represents the set of elements that are in both \(B\) and \(C\). We're asked to prove that \(A \times (B \cap C) = (A \times B) \cap (A \times C)\).

Prove the Left side Subset of Right side

The challenge here is to prove this equality by showing that each side of the equation is a subset of the other. Let us take an arbitary element (a, b) from the left side i.e, \(A \times (B \cap C)\). By definition of the Cartesian product and intersection, \(a \in A\) and \(b \in B \cap C\). Hence, \(b\) is in both \(B\) and \(C\). Therefore, \((a, b) \in A \times B\) and \((a, b) \in A \times C\), so \((a, b) \in (A \times B) \cap (A \times C)\). Therefore, \(A \times (B \cap C)\) is a subset of \((A \times B) \cap (A \times C)\).

Prove the Right side Subset of Left side

Now take an arbitrary element \((a, b)\) from the right side i.e, \((A \times B) \cap (A \times C)\). This implies that \((a, b) \in A \times B\) and \((a, b) \in A \times C\). Therefore, \(a \in A\) and \(b \in B\) and \(b \in C\). This means \(b \in B \cap C\). Hence, \((a, b) \in A \times (B \cap C)\). Therefore, \((A \times B) \cap (A \times C)\) is a subset of \(A \times (B \cap C)\).