Alright, let's go through each part of the question step by step.
### (a) Write the formula to find the number of proper subsets.
For a set [tex]\( A \)[/tex] with [tex]\( n \)[/tex] elements, the number of proper subsets is calculated using the formula:
[tex]\[ 2^n - 1 \][/tex]
This formula comes from the fact that a set with [tex]\( n \)[/tex] elements has [tex]\( 2^n \)[/tex] total subsets (including the empty set and the set itself). Proper subsets exclude the set itself, hence the subtraction of 1.
### (b) Find the number of proper subsets.
Given the set [tex]\( A = \{p, e, n\} \)[/tex]:
1. First, determine the number of elements in set [tex]\( A \)[/tex]. The set [tex]\( A \)[/tex] has 3 elements.
2. Using the formula [tex]\( 2^n - 1 \)[/tex], where [tex]\( n \)[/tex] is 3:
[tex]\[ 2^3 - 1 = 8 - 1 = 7 \][/tex]
Therefore, the number of proper subsets of set [tex]\( A \)[/tex] is 7.
### (c) Write any 2 subsets of set [tex]\( A \)[/tex] containing two elements only.
A subset of a set contains some or all elements of the set without repitition and the order of elements does not matter.
Given the set [tex]\( A = \{p, e, n\} \)[/tex], we can choose subsets containing exactly two elements:
1. Subset containing 'p' and 'e': [tex]\(\{p, e\}\)[/tex]
2. Subset containing 'e' and 'n': [tex]\(\{e, n\}\)[/tex]
These two subsets are part of the proper subsets of set [tex]\( A \)[/tex].
### Summary
1. The formula to find the number of proper subsets of a set with [tex]\( n \)[/tex] elements is [tex]\( 2^n - 1 \)[/tex].
2. The number of proper subsets of [tex]\( A = \{p, e, n\} \)[/tex] is 7.
3. Two subsets of [tex]\( A \)[/tex] containing two elements each are [tex]\(\{p, e\}\)[/tex] and [tex]\(\{e, n\}\)[/tex].
