Question

How many subsets can be formed from a set of n elements? How many of these willl be proper and
how many improper ?​

Answer 1

Answer:

If a set has “n” elements, then the number of subset of the given set is 2n and the number of proper subsets of the given subset is given by 2n-1.

Answer 2

2ⁿ subsets can be formed from a set of n elements and 2ⁿ - 1 subsets will be proper and 1 subset improper

Proper subsets of a set consists of  elements of Set but not all the elements

Improper Subset of a set is set itself ( consisting of all the elements of set)

A set of Elements of n

Subsets can be formed by selecting 0 , 1 , 2 , ...  , n elements

0 elements can be selected in ⁿC₀ ways

1 elements can be selected in ⁿC₁ ways

2 elements can be selected in ⁿC₂ ways

and so on

n elements can be selected in ⁿCₙ ways

Hence Total subsets

=  ⁿC₀ + ⁿC₁ + ⁿC₂  + ... + ⁿCₙ

= 2ⁿ

Proper subsets  is only one  as ⁿCⁿ = 1

Rest all are improper subsets

Hence 2ⁿ - 1