Let `A={1,2),B={1,2,3,4},C={5,6}` and `D={5,6,7,8}` verify that
`AxxC` is a subset of `BxxD`.

Answer

To verify that \( A \times C \) is a subset of \( B \times D \), we will follow these steps:

### Step 1: Define the sets
We have the following sets:
- \( A = \{1, 2\} \)
- \( B = \{1, 2, 3, 4\} \)
- \( C = \{5, 6\} \)
- \( D = \{5, 6, 7, 8\} \)

### Step 2: Compute \( A \times C \)
The Cartesian product \( A \times C \) consists of all ordered pairs \( (a, c) \) where \( a \in A \) and \( c \in C \).

Calculating \( A \times C \):
\[ A \times C = \{(1, 5), (1, 6), (2, 5), (2, 6)\} \]

### Step 3: Compute \( B \times D \)
The Cartesian product \( B \times D \) consists of all ordered pairs \( (b, d) \) where \( b \in B \) and \( d \in D \).

Calculating \( B \times D \):
\[ B \times D = \{(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)\} \]

### Step 4: Verify if \( A \times C \subseteq B \times D \)
We need to check if every element of \( A \times C \) is also in \( B \times D \).

- Check \( (1, 5) \): Present in \( B \times D \)
- Check \( (1, 6) \): Present in \( B \times D \)
- Check \( (2, 5) \): Present in \( B \times D \)
- Check \( (2, 6) \): Present in \( B \times D \)

Since all elements of \( A \times C \) are found in \( B \times D \), we conclude that:
\[ A \times C \subseteq B \times D \]

### Conclusion
Thus, we have verified that \( A \times C \) is indeed a subset of \( B \times D \).

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