Problem 18 For all sets $A, B$, and $C$... [FREE SOLUTION]

Chapter 5: Problem 18

For all sets $A, B$, and $C$ $$ A \times(B \cap C)=(A \times B) \cap(A \times C) $$

Short Answer

Expert verified

To prove that $A \times(B \cap C)=(A \times B) \cap(A \times C)$, we will show that $A \times(B \cap C) \subseteq (A \times B) \cap(A \times C)$ and $(A \times B) \cap(A \times C) \subseteq A \times(B \cap C)$. We do this by demonstrating that for any arbitrary element (x, y) in either side of the equation, it must be an element in the other side as well. After verifying the subset relationships in both directions, we can conclude that $A \times(B \cap C)=(A \times B) \cap(A \times C)$.

Step by step solution

Proving that $A \times(B \cap C) \subseteq (A \times B) \cap(A \times C)$

Let (x, y) be an arbitrary element in $A \times(B \cap C)$. By definition, this means that x is in A and y is in the intersection of B and C, i.e., y is in both B and C. Now, since x is in A and y is in B, we can say that (x, y) is in $A \times B$. Similarly, since x is in A and y is in C, we can also say that (x, y) is in $A \times C$. Hence, (x, y) is in both $A \times B$ and $A \times C$, meaning $(x, y) \in (A \times B) \cap(A \times C)$. Since (x, y) was arbitrary, this holds for all elements in $A \times(B \cap C)$, proving that $A \times(B \cap C) \subseteq (A \times B) \cap(A \times C)$.

Proving that $(A \times B) \cap(A \times C) \subseteq A \times(B \cap C)$

Let (x, y) be an arbitrary element in $(A \times B) \cap(A \times C)$. By definition, this means that (x, y) is in both $A \times B$ and $A \times C$. Since (x, y) is in $A \times B$, we know that x is in A and y is in B. Similarly, since (x, y) is in $A \times C$, we know that x is in A and y is in C. Therefore, x is in A, and y is in both B and C. This implies that y is in the intersection of B and C, i.e., y is in $B \cap C$. We can now say that (x, y) is in $A \times(B \cap C)$. Since (x, y) was arbitrary, this holds for all elements in $(A \times B) \cap(A \times C)$, proving that $(A \times B) \cap(A \times C) \subseteq A \times(B \cap C)$.

Concluding equality

From Step 1 and Step 2, we have shown that $A \times(B \cap C) \subseteq (A \times B) \cap(A \times C)$ and $(A \times B) \cap(A \times C) \subseteq A \times(B \cap C)$. Since there is a subset relationship in both directions, we can conclude that $A \times(B \cap C)=(A \times B) \cap(A \times C)$.

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For all sets \(A, B\), and \(C\) $$ A \times(B \cap C)=(A \times B) \cap(A \times C) $$

Short Answer

Step by step solution

Proving that \(A \times(B \cap C) \subseteq (A \times B) \cap(A \times C)\)

Proving that \((A \times B) \cap(A \times C) \subseteq A \times(B \cap C)\)

Concluding equality

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