Question

Use Algorithm 2 to list all the subsets of the set\({\rm{\{ 1,2,3,4\} }}\).

Short answer

All the subsets of set\({\rm{\{ 1,2,3,4\} }}\)is:

\(\left\{ {\begin{array}{*{20}{l}}{\phi ,(1),(2),(3),(4),(1,2),(1,3),(1,4),(2,3)(2,4)}\\{(3,4),(1,2,3),(2,3,4),(1,2,4),(1,3,4)(1,2,3,4)}\end{array}} \right\}\)

Step-by-step solution

Definition of Concept

Permutations: A permutation of a set is a loosely defined arrangement of its members into a sequence or linear order, or, if the set is already ordered, a rearrangement of its elements, in mathematics. The act of changing the linear order of an ordered set is also referred to as "permutation."

Lexicographic order: The lexicographic or lexicographical order (also known as lexical order or dictionary order) in mathematics is a generalisation of the alphabetical order of dictionaries to sequences of ordered symbols or, more broadly, elements of a totally ordered set.

List all the subsets of the given set using Algorithm 2

Considering the given information:

The set\({\rm{\{ 1,2,3,4\} }}\).

Using the following concept:

A subset of a given set is essentially a collection of items that belong to that set but are conditionally related to one another.

Algorithm

Because a set contains four different integers, it can be represented by a four-bit string.

Let 0 denote that it is devoid of elements, and 1 denote that it contains elements.

As a result, here is a list of all possible bit strings:

\({\rm{\{ 0000,1000,0100,0010,0001,1100,1010,0110,0101,0011,1110,0111,1101,1011,1111\} }}\)

Below is a list of possible subsets for the above-mentioned set.

\(\left\{ {\begin{array}{*{20}{l}}{\phi ,(1),(2),(3),(4),(1,2),(1,3),(1,4),(2,3)(2,4)}\\{(3,4),(1,2,3),(2,3,4),(1,2,4),(1,3,4)(1,2,3,4)}\end{array}} \right\}\)

Therefore, the required all the subsets of set\({\rm{\{ 1,2,3,4\} }}\)is:

\(\left\{ {\begin{array}{*{20}{l}}{\phi ,(1),(2),(3),(4),(1,2),(1,3),(1,4),(2,3)(2,4)}\\{(3,4),(1,2,3),(2,3,4),(1,2,4),(1,3,4)(1,2,3,4)}\end{array}} \right\}\)