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Sequence

A sequence is a fundamental concept in formal order analysis that is defined as a finite, enumerated collection of objects in which repetitions are allowed and order matters. It is the basic object for analyzing patterns, relationships, and structural properties in ordered data.

Mathematical Definition

A sequence \(S\) is as tuple:

\[S = <s_1, s_2, ..., s_n>\]

where:

  • \(n\) is the length of the sequence
  • \(s_i \in X\) for all \(i \in \{1,2,...,n\}\)
  • \(X\) is an unordered set
  • \(=\) is the equivalence relation defined on \(X\)

Examples

Binary Sequence

A binary sequence 0110100110

represented as

\(X = \{0,1\}\)

\(S = <0,1,1,0,1,0,0,1,1,0>\)

Musical Chorus Sequence

A musical chorus for Jingle bell rock

D                Dmaj7        D6
Jingle-bell, Jingle-bell, Jingle-bell Rock.
  D                D#dim
Jingle-bell swing and
 Em           A7     Em               A7            Em A7
Jingle-bell ring. Snowin' and blowin' up bushels of fun.
Em  A9                  A7
Now the jingle-hop has begun.

\(X = \{D, D6, D\#dim, A7, A9, Dmaj7, Em\}\) \(S = <D,Dmaj7,D6,D,D\#dim,Em,A7,Em,A7,Em,A7,Em,A9,A7>\)

DNA Sequence

A DNA sequence ATGCTAGCATGCTAGCATGCTAGC

\(X = \{A,C,T,G\}\) \(S = <A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C,A,T,G,C,T,A,G,C>\)

English Text Sequence as char sequence

An English text sentence the quick brown fox jumps over the lazy dog

\(X = \{\ ,a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\}\) \(S = <t,h,e,\ ,q,u,i,c,k,\ ,b,r,o,w,n,\ ,f,o,x,\ ,j,u,m,p,s,\ ,o,v,e,r,\ ,t,h,e,\ ,l,a,z,y,\ ,d,o,g>\)

English Text Sequence as word sequence

An English text sentence the quick brown fox jumps over the lazy dog

\(X = \{\ ,quick, fox, brown, the, over, dog, fox, lazy\}\) \(S = <the,\ ,quick,\ ,brown,\ ,fox,\ ,jumps,\ ,over,\ ,the,\ ,lazy,\ ,dog>\)