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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Let S be a set and P(S) be the set of all subsets of S. Show that S is “smaller than” P(S) in the sense that there is a one-to-one function from S to P(S) but there is no onto function from S to P(S).

To determine

To prove:

S is smaller than P(S) such that there is a one-to-one function from S to P(S) but there is no onto function from S to P(S).

Explanation

Given information:

S be a set and let P(S) be the set of all subsets of S.

Concept used:

P(S) be the power set.

Proof:

Consider, S be a set and P(S) be the set of all S.

Objective is to show that S is smaller than P(S).

For this show that there is a one-to-one map from S to P(S) but there is no onto function from P(S) to S so, define a map f:SP(S) as f(x)={x}, for all xS.

Then clearly, f(x)=f(y) implies that {x}={y}. Since the sets are singleton therefore, this further implies that x=y

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