The principal objects of interest in the current research are the finite sets and the contraction finite transformation semigroups and the characterization of nildempotent elements in . Let be a finite set, say = { , , … }, where is a non-negative integer then ∈ for which for all , ∈ , | − | ≤ | − | is a contraction mapping for all , ∈ , provided that any element in ( ) is not assumed to be mapped to empty as a contraction. We show that ∈ is nildempotent if there exist some minimal (nildempotent degree) , ∈ such that = ∅ ⟹ = where = then = = = ∅ implies ⊆ | ( )| where = for each ∈ . Then = − + , , ∈ for 1≥ ≥ .
Mathematics of music and sound production brings to bear the physical and practical application of Mathematics in the field of Music. The composition of songs involves the principle of key scaling, their respective interval as calculated with the aid of an appropriate key division which suites the generality of songs composed in different keys with the keyboard. Melodies, harmonies and rhythms produced in the stage of rendition is characterized by transposition and inversion of key to suite each song. With the aid of keyboard, elements of Symmetric group are used to compose songs.
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