To solve the problem of finding the set of ordered pairs in the relation "is greater than" from [tex]\( \Delta \)[/tex] to [tex]\( B \)[/tex], let's proceed step-by-step and determine the ordered pairs [tex]\((a, b)\)[/tex] such that [tex]\( a \in \Delta \)[/tex] and [tex]\( b \in B \)[/tex] where [tex]\( a \)[/tex] is greater than [tex]\( b \)[/tex].
Given:
[tex]\[ \Delta = \{1, 2, 3\} \][/tex]
[tex]\[ B = \{1, 3, 5\} \][/tex]
We need to compare each element of [tex]\(\Delta\)[/tex] with each element of [tex]\(B\)[/tex] to identify pairs where the element from [tex]\(\Delta\)[/tex] is greater than the element from [tex]\(B\)[/tex].
1. Let's start with [tex]\(1 \in \Delta\)[/tex]:
- Compare [tex]\(1\)[/tex] with each element in [tex]\(B\)[/tex]:
- [tex]\(1 > 1\)[/tex]? No.
- [tex]\(1 > 3\)[/tex]? No.
- [tex]\(1 > 5\)[/tex]? No.
There are no ordered pairs for [tex]\(1 \in \Delta\)[/tex].
2. Next, let's consider [tex]\(2 \in \Delta\)[/tex]:
- Compare [tex]\(2\)[/tex] with each element in [tex]\(B\)[/tex]:
- [tex]\(2 > 1\)[/tex]? Yes, so [tex]\((2, 1)\)[/tex] is an ordered pair.
- [tex]\(2 > 3\)[/tex]? No.
- [tex]\(2 > 5\)[/tex]? No.
The ordered pair for [tex]\(2 \in \Delta\)[/tex] is [tex]\((2, 1)\)[/tex].
3. Finally, let's look at [tex]\(3 \in \Delta\)[/tex]:
- Compare [tex]\(3\)[/tex] with each element in [tex]\(B\)[/tex]:
- [tex]\(3 > 1\)[/tex]? Yes, so [tex]\((3, 1)\)[/tex] is an ordered pair.
- [tex]\(3 > 3\)[/tex]? No.
- [tex]\(3 > 5\)[/tex]? No.
The ordered pair for [tex]\(3 \in \Delta\)[/tex] is [tex]\((3, 1)\)[/tex].
Combining all the results, the set of ordered pairs [tex]\((a, b)\)[/tex] in the relation "is greater than" from [tex]\(\Delta\)[/tex] to [tex]\(B\)[/tex] is:
[tex]\[ \{(2, 1), (3, 1)\} \][/tex]
Thus, the set of ordered pairs in the relation "is greater than" from [tex]\(\Delta\)[/tex] to [tex]\(B\)[/tex] is:
[tex]\[ \{(3, 1), (2, 1)\} \][/tex]
