Certainly! Let's walk through the problem step by step.
Given the set [tex]\( A = \{8, 9, 10, 11, 12, 14\} \)[/tex]:
### Part (a): Number of Subsets
To determine how many subsets a set has, we use the formula for the number of subsets of a set. For a set with [tex]\( n \)[/tex] elements, the number of subsets is given by [tex]\( 2^n \)[/tex].
1. First, we determine the number of elements in set [tex]\( A \)[/tex].
[tex]\[
|A| = 6
\][/tex]
since there are six elements in the set [tex]\( A \)[/tex].
2. Next, we apply the formula [tex]\( 2^n \)[/tex] to find the number of subsets.
[tex]\[
2^6 = 64
\][/tex]
Thus, the number of subsets of the set [tex]\( A \)[/tex] is [tex]\( 64 \)[/tex].
### Part (b): Number of Proper Subsets
A proper subset of a set is a subset that is not equal to the original set. In other words, it excludes the set itself.
1. The total number of subsets of set [tex]\( A \)[/tex] is [tex]\( 64 \)[/tex] (as calculated in part (a)).
2. To find the number of proper subsets, we subtract 1 from the total number of subsets because the only subset that is not a proper subset is the set itself.
[tex]\[
64 - 1 = 63
\][/tex]
Thus, the number of proper subsets of the set [tex]\( A \)[/tex] is [tex]\( 63 \)[/tex].
### Summary:
- The number of subsets of the set [tex]\( A \)[/tex] is [tex]\( 64 \)[/tex].
- The number of proper subsets of the set [tex]\( A \)[/tex] is [tex]\( 63 \)[/tex].
