Question

Describe a procedure for listing all the subsets of a finite set.

Short answer

First add the empty set, then the set containing one element of the set S, then the set containing exactly two elements of the set S and so on until the set itself is added.

Step-by-step solution

The empty set

The empty set does not contain elements nor sets.

It is denoted by\(\phi \).

Subset:

X is a subset of Y if every element of X is also an element of Y.

It is denoted by \(X \subseteq Y\)

To describe a process for listing all the subsets of a finite set.

Assume that finite set S contains n elements.

Let A be the listing of all subsets of S.

The first set is an empty set \(\phi \) that is added to the listing of A.

\(\phi \in A\)

Then, add all subsets containing one element of the set S.

\(\forall x \in S:\left\{ x \right\} \in A\)

Then, add all subsets containing exactly two elements of the set S.

\(\forall x \in S \wedge \forall y \in S \wedge x \ne y:\left\{ {x,y} \right\} \in A\)

And so on add the subset containing exactly n elements of the set S which is S itself.

\(S \in A\)

Thus, first add the empty set, then the set containing one element of the set S, then the set containing exactly two elements of the set S and so on until the set itself is added.