Math, asked by menisharamsiej, 6 months ago

If A be a finite set having n element,then there are 2n distinct subset of A?​

Answers

Answered by jannani143
0

Answer:

this is the correct answer

Step-by-step explanation:

Prove: For any finite set S,  if |S|= n, then S has 2n subsets.

Proof by induction.

Let P(n) be the predicate  “A set with cardinality n has 2n subsets.

Basis step:

P(0) is true, because the set with cardinality 0 (the empty set) has 1 subset (itself)  

and 20 = 1.

Inductive step

Prove P(k) → P(k+1)

That is, prove that if a set with k elements has 2k subsets, then a set with k+1  

elements has 2k+1 subsets.

Proof

Assume that for an arbitrary k, any set with cardinality k has 2k subsets.

Let T be a set such that |T| = k+1.

Enumerate the elements of T:  

T = e1,e2,...ek,e { k+1}.

Let  

S = e1,e2,...e { k}

Then |S|=k, so S has 2k subsets, according to the inductive hypothesis.

Note that  

T = S ∪ e{ k+1}, so every subset of S is also a subset of T.

Any subset of T either contains the element

€ ek+1, or it doesn’t contain  

€ ek+1.

If a subset of T doesn’t contain

€ ek+1, then it is also a subset of S, and there are 2k of  

those subsets. On the other hand, if a subset of T does contain the element  € ek+1, then that subset is  formed by including  € ek+1 in one of the 2k subsets of S, so T has 2k subsets containing€

ek+1.

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