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

Short Answer

Expert verified

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

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

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01

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.

02

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:

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

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Most popular questions from this chapter

How many subsets with more than two elements does a set with 100elements have?

a) Let nand rbe positive integers. Explain why the number of solutions of the equationx1+x2+...+xn=r,wherexiis a nonnegative integer forrole="math" localid="1668688407359" i=1,2,3,....,n,equals the number of r-combinations of a set with nelements.

b) How many solutions in nonnegative integers are there to the equationrole="math" localid="1668688467718" x1+x2+x3+x4=17?

c) How many solutions in positive integers are there to the equation in part (b)?

x101y99What is the coefficient ofin the expansion of(2xโˆ’3y)200?

How many bit strings of length \({\rm{10}}\) over the alphabet \({\rm{\{ a,b,c\} }}\) have either exactly three \({\rm{a}}\)s or exactly four \({\rm{b}}\)s?

One hundred tickets, numbered \(1,2,3, \ldots ,100\), are sold to \(100\) different people for a drawing. Four different prizes are awarded, including a grand prize (a trip to Tahiti). How many ways are there to award the prizes if

a) there are no restrictions?

b) the person holding ticket \(47\) wins the grand prize?

c) the person holding ticket \(47\) wins one of the prizes?

d) the person holding ticket \(47\) does not win a prize?

e) the people holding tickets \(19\) and \(47\) both win prizes?

f) the people holding tickets \(19\;,\;47\)and \(73\) all win prizes?

g) the people holding tickets \(19\;,\;47\;,\;73\) and \(97\) all win prizes?

h) none of the people holding tickets \(19\;,\;47\;,\;73\) and \(97\) wins a prize?

i) the grand prize winner is a person holding ticket \(19\;,\;47\;,\;73\) or \(97\)?

j) the people holding tickets 19 and 47 win prizes, but the people holding tickets \(73\) and \(97\) do not win prizes?

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