Sure, let's work through the problem step-by-step.
Given sets:
- [tex]\( A = \{36\} \)[/tex]
- [tex]\( B = \{1, 2, 3\} \)[/tex]
Let's find the Cartesian products [tex]\( A \times B \)[/tex] and [tex]\( B \times A \)[/tex].
### a) Cartesian Product [tex]\( A \times B \)[/tex]
The Cartesian product [tex]\( A \times B \)[/tex] is the set of all ordered pairs [tex]\((a, b)\)[/tex] where [tex]\( a \)[/tex] is an element of [tex]\( A \)[/tex] and [tex]\( b \)[/tex] is an element of [tex]\( B \)[/tex].
Since [tex]\( A \)[/tex] contains only one element, 36, we pair it with each of the elements of [tex]\( B \)[/tex]:
1. Pairing 36 (from [tex]\( A \)[/tex]) with 1 (from [tex]\( B \)[/tex]) gives us the ordered pair [tex]\((36, 1)\)[/tex].
2. Pairing 36 (from [tex]\( A \)[/tex]) with 2 (from [tex]\( B \)[/tex]) gives us the ordered pair [tex]\((36, 2)\)[/tex].
3. Pairing 36 (from [tex]\( A \)[/tex]) with 3 (from [tex]\( B \)[/tex]) gives us the ordered pair [tex]\((36, 3)\)[/tex].
Therefore,
[tex]\[ A \times B = \{ (36, 1), (36, 2), (36, 3) \} \][/tex]
### b) Cartesian Product [tex]\( B \times A \)[/tex]
The Cartesian product [tex]\( B \times A \)[/tex] is the set of all ordered pairs [tex]\((b, a)\)[/tex] where [tex]\( b \)[/tex] is an element of [tex]\( B \)[/tex] and [tex]\( a \)[/tex] is an element of [tex]\( A \)[/tex].
Since [tex]\( A \)[/tex] contains only one element, 36, we pair each element of [tex]\( B \)[/tex] with 36 from [tex]\( A \)[/tex]:
1. Pairing 1 (from [tex]\( B \)[/tex]) with 36 (from [tex]\( A \)[/tex]) gives us the ordered pair [tex]\((1, 36)\)[/tex].
2. Pairing 2 (from [tex]\( B \)[/tex]) with 36 (from [tex]\( A \)[/tex]) gives us the ordered pair [tex]\((2, 36)\)[/tex].
3. Pairing 3 (from [tex]\( B \)[/tex]) with 36 (from [tex]\( A \)[/tex]) gives us the ordered pair [tex]\((3, 36)\)[/tex].
Therefore,
[tex]\[ B \times A = \{ (1, 36), (2, 36), (3, 36) \} \][/tex]
### Representing on the Cartesian Plane:
For part a) [tex]\( A \times B \)[/tex]:
- Points on Cartesian Plane: (36, 1), (36, 2), (36, 3)
For part b) [tex]\( B \times A \)[/tex]:
- Points on Cartesian Plane: (1, 36), (2, 36), (3, 36)
These ordered pairs are the coordinates that will be plotted on the Cartesian plane for each respective product.
