To determine the Cartesian product [tex]\( A \times B \)[/tex] where [tex]\( A = \{2, 5, 7, 8\} \)[/tex] and [tex]\( B = \{1, 5, 7\} \)[/tex], we follow these steps:
1. The Cartesian product [tex]\( A \times B \)[/tex] is defined as the set of all ordered pairs [tex]\((a, b)\)[/tex] where [tex]\( a \in A \)[/tex] and [tex]\( b \in B \)[/tex].
2. We produce each ordered pair [tex]\((a, b)\)[/tex] by taking each element [tex]\( a \)[/tex] from set [tex]\( A \)[/tex] and pairing it with each element [tex]\( b \)[/tex] from set [tex]\( B \)[/tex].
Let’s proceed with the calculations step by step as follows:
- For [tex]\( a = 2 \)[/tex]:
- Pair with [tex]\( b = 1 \)[/tex] to get [tex]\((2, 1)\)[/tex].
- Pair with [tex]\( b = 5 \)[/tex] to get [tex]\((2, 5)\)[/tex].
- Pair with [tex]\( b = 7 \)[/tex] to get [tex]\((2, 7)\)[/tex].
- For [tex]\( a = 5 \)[/tex]:
- Pair with [tex]\( b = 1 \)[/tex] to get [tex]\((5, 1)\)[/tex].
- Pair with [tex]\( b = 5 \)[/tex] to get [tex]\((5, 5)\)[/tex].
- Pair with [tex]\( b = 7 \)[/tex] to get [tex]\((5, 7)\)[/tex].
- For [tex]\( a = 7 \)[/tex]:
- Pair with [tex]\( b = 1 \)[/tex] to get [tex]\((7, 1)\)[/tex].
- Pair with [tex]\( b = 7 \)[/tex] to get [tex]\((7, 5)\)[/tex].
- Pair with [tex]\( b = 8 \)[/tex] to get [tex]\((7, 7)\)[/tex].
- For [tex]\( a = 8 \)[/tex]:
- Pair with [tex]\( b = 1 \)[/tex] to get [tex]\((8, 1)\)[/tex].
- Pair with [tex]\( b = 5 \)[/tex] to get [tex]\((8, 5)\)[/tex].
- Pair with [tex]\( b = 7 \)[/tex] to get [tex]\((8, 7)\)[/tex].
Combining all these ordered pairs, we get:
[tex]\[
A \times B = \{(2, 1), (2, 5), (2, 7), (5, 1), (5, 5), (5, 7), (7, 1), (7, 5), (7, 7), (8, 1), (8, 5), (8, 7)\}
\][/tex]
So, the Cartesian product [tex]\( A \times B \)[/tex] is:
[tex]\[
\{(2, 1), (2, 5), (2, 7), (5, 1), (5, 5), (5, 7), (7, 1), (7, 5), (7, 7), (8, 1), (8, 5), (8, 7)\}.
\][/tex]
This is the complete set of ordered pairs that forms the Cartesian product of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
