To begin solving the given problem, we need to address the set notation provided:
[tex]\[ \Delta:\{x+2 \mid x \in \mathbb{N}, 2 < x < 10 \} \][/tex]
Let's break this down step-by-step.
1. Understanding the Set Notation:
- [tex]\( \mathbb{N} \)[/tex] denotes the set of natural numbers, typically starting from 1.
- The condition [tex]\( 2 < x < 10 \)[/tex] indicates we are interested in natural numbers [tex]\( x \)[/tex] that lie strictly between 2 and 10.
2. Identifying the Range of [tex]\( x \)[/tex]:
The numbers that satisfy [tex]\( 2 < x < 10 \)[/tex] are:
[tex]\[ x \in \{3, 4, 5, 6, 7, 8, 9 \} \][/tex]
This means the natural numbers in the specified range are 3, 4, 5, 6, 7, 8, and 9.
3. Applying the Transformation [tex]\( x + 2 \)[/tex]:
- For [tex]\( x = 3 \)[/tex]:
[tex]\[ x + 2 = 3 + 2 = 5 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ x + 2 = 4 + 2 = 6 \][/tex]
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ x + 2 = 5 + 2 = 7 \][/tex]
- For [tex]\( x = 6 \)[/tex]:
[tex]\[ x + 2 = 6 + 2 = 8 \][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ x + 2 = 7 + 2 = 9 \][/tex]
- For [tex]\( x = 8 \)[/tex]:
[tex]\[ x + 2 = 8 + 2 = 10 \][/tex]
- For [tex]\( x = 9 \)[/tex]:
[tex]\[ x + 2 = 9 + 2 = 11 \][/tex]
4. Constructing the Resulting Set [tex]\( \Delta \)[/tex]:
By collecting all the results from the transformation [tex]\( x + 2 \)[/tex], we get:
[tex]\[ \Delta = \{5, 6, 7, 8, 9, 10, 11\} \][/tex]
Thus, the final result for the set [tex]\( \Delta \)[/tex] is:
[tex]\[ \Delta = \{5, 6, 7, 8, 9, 10, 11\} \][/tex]
This is the comprehensive breakdown of the solution for the given problem.
