To find the complement of the given set within the specified universal set, we follow these steps:
1. Define the Universal Set [tex]\( U \)[/tex]:
The universal set [tex]\( U \)[/tex] is the set of all integers from -3 to 6, inclusive. This can be written as:
[tex]\[
U = \{-3, -2, -1, 0, 1, 2, 3, 4, 5, 6\}
\][/tex]
2. Define Set [tex]\( A \)[/tex]:
The given set [tex]\( A \)[/tex] includes all integers from -2 to less than 2. This can be written as:
[tex]\[
A = \{-2, -1, 0, 1\}
\][/tex]
3. Determine the Complement of Set [tex]\( A \)[/tex] in [tex]\( U \)[/tex]:
The complement of set [tex]\( A \)[/tex] within the universal set [tex]\( U \)[/tex] is the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex]. Thus, we need to list all elements from [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex].
4. Identify the Elements:
- [tex]\( U \)[/tex] contains the elements [tex]\(-3, -2, -1, 0, 1, 2, 3, 4, 5, 6 \)[/tex].
- [tex]\( A \)[/tex] contains the elements [tex]\(-2, -1, 0, 1 \)[/tex].
By excluding the elements of [tex]\( A \)[/tex] from [tex]\( U \)[/tex], we get:
[tex]\[
U \setminus A = \{-3, 2, 3, 4, 5, 6\}
\][/tex]
5. List the Result:
The elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex] (i.e., the complement of [tex]\( A \)[/tex] in [tex]\( U \)[/tex]) are:
[tex]\[
\{-3, 2, 3, 4, 5, 6\}
\][/tex]
Therefore, the complement of the set [tex]\( \{x \mid x \in I \text{ and } -2 \leq x < 2\} \)[/tex] with respect to the universal set [tex]\( \{x \mid x \in I \text{ and } -3 \leq x \leq 6\} \)[/tex] is:
[tex]\[
\{-3, 2, 3, 4, 5, 6\}
\][/tex]
So, the final answer is:
[tex]\[
-3, 2, 3, 4, 5, 6
\][/tex]
