Math, asked by dikshyadhungana3, 10 months ago

A = {1, 2, 3} the number of non-empty proper subsets are

Answers

Answered by shampa107roy

Answer:

Step-by-step explanation:

Let us take an arbitrary Set S which is a finite set and has cardinality |S|=n

So then the set S will be having 2n subsets. This can be easily proven -

For every element in the set S, it has 2 choices

It is included in the subset

It is not included in the subsets

Let S={s1,s2,s3⋯sn}

So the choices will be 2×2×⋯×2(ntimes)

The empty subset will be the case where we would have not choose any element and it is 1 case and the case in which we have choose all elements will be the set itself

Now the proper subset of a set is the subset in which the cardinality of the that subset is less than the set itself.

So the there are two subsets which do not satisfy the above conditions. They are empty subset itself and the subset which is the set itself.

So proper subsets which are non empty are = 2n−2=2(2n−1)

Now coming onto your question, set A={1,2,3,4}

All subsets are also called powerset.

P(A)=

{

ϕ,

{1},

{2},

{3},

{4},

{1,2},

{1,3},

{1,4},

{2,3},

{2,4},

{3,4},

{1,2,3},

{1,2,4}

{1,3,4},

{2,3,4}

{1,2,3,4}

}

if we eliminate the first and last subsets , we will get the required answer.

Hence the answer is