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Alphabet

An alphabet is a m-tuple of unique elements.

Mathematical Definition

Let \(X\) is Carrier set

The alphabet \(A\) is a m-tuple with a uniqueness constraint, can be defined:

\[A = <a_1, a_2, ..., a_m>,\]
\[\forall i,j \in \{1, ... ,m\}, i \neq j \implies a_i \neq a_j,\]
\[\forall i \in \{1, ... ,m\} \exists a_i \in X \]

Where:

  • \(m := |A|\) is called power of the alphabet, \(m \in N\)
  • \(a_i\)​ is called the \(i\)-th element (or coordinate) of the alphabet.

Alphabet of Sequence

Let \(X\) is Carrier set

Let \(S\) is Sequence described as function \(S : \{1,...,n\} \longrightarrow X\)

\[alphabet(S) : \big\{\{1,...,n\} \longrightarrow X \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}\]
\[alphabet(S) = \big<S(i) \big| i \in \{1,...,n\}, \forall k < i, S(i) \neq S(k)\big>\]

Where:

  • \(m \leq n\) - power of the alphabet is not greater than length of the sequence

Examples

Binary Sequence

A binary sequence 0110100110

represented as

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

\(A = <0,1>\)

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.

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

\(A = <D,Dmaj7,D6,D,D\#dim,Em,A7,A9>\)

DNA Sequence

A DNA sequence ATGCTAGCATGCTAGCATGCTAGC

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

\(A = <A,T,G,C>\)

English Text Sequence as char sequence

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

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

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

English Text Sequence as word sequence

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

\(X = \{\ ,quick, fox, brown, the, over, dog, fox, lazy\}\)

\(A = <the,\ ,quick,brown,fox,jumps,over,lazy,dog>\)