To determine the relative complement of [tex]\( A \)[/tex] with respect to [tex]\( B \)[/tex], we need to find the elements that are in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex].
Given sets:
[tex]\[ A = \{-1, \{0\}, \{-1, 0\}\} \][/tex]
[tex]\[ B = \{\{-1\}, -1, 0\} \][/tex]
Let's analyze which elements of [tex]\( B \)[/tex] are not in [tex]\( A \)[/tex] step-by-step:
1. Element analysis from [tex]\( B \)[/tex]:
- The element [tex]\( 0 \)[/tex] is in [tex]\( B \)[/tex]. We need to see if [tex]\( 0 \)[/tex] is in [tex]\( A \)[/tex]:
Since [tex]\( A \)[/tex] contains [tex]\(-1\)[/tex], \{\{0\}\}, and \{\{-1, 0\}\}, but does not contain [tex]\( 0 \)[/tex] by itself, [tex]\( 0 \)[/tex] is in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex].
- The element \{-1\} (which is a set containing [tex]\(-1\)[/tex]) is in [tex]\( B \)[/tex]. We need to see if [tex]\( \{-1\} \)[/tex] is in [tex]\( A \)[/tex]:
[tex]\( A \)[/tex] does not contain \{-1\} (it only contains the individual element [tex]\(-1\)[/tex]), so \{-1\} is also in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex].
2. Additional elements:
- The element [tex]\(-1\)[/tex] is in both [tex]\( A \)[/tex] and [tex]\( B \)[/tex], so it is not included in the relative complement since we are looking for elements in [tex]\( B \)[/tex] that are not in [tex]\( A \)[/tex].
By compiling our findings:
- The elements which are in [tex]\( B \)[/tex] but not in [tex]\( A \)[/tex] are [tex]\( 0 \)[/tex] and \{-1\}.
Hence, the relative complement of [tex]\( A \)[/tex] with respect to [tex]\( B \)[/tex] is:
[tex]\[ \{0, \{-1\}\} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\{0, \{-1\}\}} \][/tex]
So option A, [tex]\(\{\{-1\}, 0\}\)[/tex], is correct.
