Let's break this problem into two parts and solve it step by step.
Part 1: Finding the complement of set [tex]\( B \)[/tex] in set [tex]\( U \)[/tex].
Given:
- Set [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- Set [tex]\( B = \{2, 3, 4\} \)[/tex]
The complement of [tex]\( B \)[/tex] in [tex]\( U \)[/tex], denoted as [tex]\( \bar{B} \)[/tex], consists of all the elements in [tex]\( U \)[/tex] that are not in [tex]\( B \)[/tex].
To find [tex]\( \bar{B} \)[/tex], we subtract the elements of [tex]\( B \)[/tex] from [tex]\( U \)[/tex]:
[tex]\[
\bar{B} = U - B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} - \{2, 3, 4\}
\][/tex]
Removing the elements 2, 3, and 4 from [tex]\( U \)[/tex], we get:
[tex]\[
\bar{B} = \{1, 5, 6, 7, 8, 9, 10\}
\][/tex]
So, the correct answer for the complement of [tex]\( B \)[/tex] in [tex]\( U \)[/tex] is:
a) \{1, 5, 6, 7, 8, 9, 10\}
Part 2: Finding the total number of subsets of a set [tex]\( A \)[/tex] having 7 elements.
Given:
- Set [tex]\( A \)[/tex] has 7 elements.
The total number of subsets of any set is given by the formula [tex]\( 2^n \)[/tex], where [tex]\( n \)[/tex] is the number of elements in the set.
For a set [tex]\( A \)[/tex] having 7 elements:
[tex]\[
\text{Total number of subsets} = 2^7 = 128
\][/tex]
Therefore, the total number of subsets of a set [tex]\( A \)[/tex] with 7 elements is:
128
