Let's address each part of the question systematically:

Part C: Set Operations

You are given the following sets:

• $A = \{1, 2\}$
• $B = \{a, b, c\}$
• $C = \{c, d\}$

You are asked to find two things:

1. $(A \times B) \cap (A \times C)$
2. $A \times (B \cap C)$

1. Finding $(A \times B) \cap (A \times C)$

• The Cartesian product $A \times B$ is the set of all ordered pairs where the first element is from $A$ and the second element is from $B$. $A \times B = \{(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)\}$

• Similarly, the Cartesian product $A \times C$ is the set of all ordered pairs where the first element is from $A$ and the second element is from $C$. $A \times C = \{(1, c), (1, d), (2, c), (2, d)\}$

• Now, we find the intersection $(A \times B) \cap (A \times C)$, which consists of the pairs that are in both $A \times B$ and $A \times C$. $(A \times B) \cap (A \times C) = \{(1, c), (2, c)\}$

2. Finding $A \times (B \cap C)$

• The intersection $B \cap C$ consists of the elements that are in both $B$ and $C$. $B \cap C = \{c\}$

• The Cartesian product $A \times (B \cap C)$ is the set of all ordered pairs where the first element is from $A$ and the second element is from $B \cap C$. $A \times (B \cap C) = \{(1, c), (2, c)\}$

Thus, both answers are:

• $(A \times B) \cap (A \times C) = \{(1, c), (2, c)\}$
• $A \times (B \cap C) = \{(1, c), (2, c)\}$

Part D: Binary Operations

You are asked to verify whether the following operations define a binary operation on the set $R$.

A binary operation on a set $R$ is an operation that takes two elements from $R$ and produces another element that is also in $R$.

1. Multiplication in $R$

For multiplication to define a binary operation, the product of any two elements of $R$ must also be in $R$. This is true for the set of real numbers $R$, so multiplication in $R$ is a binary operation.

2. Division in $R$

For division to define a binary operation, the result of dividing any two elements of $R$ must also be in $R$. However, division by zero is undefined in $R$. Therefore, division in $R$ is not a binary operation because it is not always defined (when the divisor is zero).

3. Subtraction in $R$

For subtraction to define a binary operation, the result of subtracting any two elements of $R$ must also be in $R$. Since subtraction of any two real numbers produces another real number, subtraction in $R$ is a binary operation.

Would you like more details or further clarification on any part?

Relative Questions:

1. How is the intersection of two Cartesian products defined in general?
2. What is the significance of a binary operation in abstract algebra?
3. Why does division fail to be a binary operation in real numbers?
4. Can you generalize the concept of Cartesian products for more than two sets?
5. How does set theory relate to other fields of mathematics like topology or combinatorics?

Tip:

Always ensure to check the closure property when verifying binary operations for a given set!