Symbols, Compound symbols, Words

What is a symbol?

letters, words, sentences can all be a symbol
String of symbols like, 'binary digits' make up letters, 'letters' make up words and 'words' make up sentences. Thus any of these: letters, words, sentences could be a symbol.
Therefore, symbol selection is determined by the task intended.

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Compound Symbol

Consider two information sources, X & Y emitting symbols xi ∈ X and yj ∈ Y respectively.

compound symbols

Then, symbol cij formed by the combination of symbols from X & Y is called Compound Symbol. The compound symbol therefore is an ordered pair ⟨xi, yj⟩ taken from a set of symbols C, given by
C = set of all possible c sub ij

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Word, wi : A special case of compound symbol cij

Consider an information source, X emitting n symbols, xi ∈ X into (say) a shift-register. Thus it emits n–symbols in parallel.

word is a compound symbol from (say) a shift register

Then, symbol output is a collective parallel read out. For example, word wi.

The symbol wi therefore is an ordered pair of the symbol xi in specific positions ⟨xi,0, xi,1, xi,2, …, xi,n−1⟩ taken from a set of all possible wi symbols W is given by

W = set of all possible w sub i
Thus cardinality
| wi | ≜ n
and
| W | ≤ | X |n

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Illustration of | wi | ≜ n and | W | ≤ | X |n

Imagine X = {p, q}, thus | X | = 2 with possible words; w0 = ⟨p, q⟩, w1 = ⟨q, p⟩, w2 = ⟨p, p⟩ and, w3 = ⟨q, q⟩.
Therefore, | wi | = 2 and | W | = | X |2 = 22 = 4.
Notice that if the rule is such that no symbol in a word wi can repeat then W = {w0, w1, w2, w3}. Therefore, for this case | W | = 2. Hence, the upper bound for | W | is 4.
Hence, for the general case W | ≤ | X |2.

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Next:

Mathematics of compound symbols (p:2) ➽

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