## What is a symbol?

*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.**❶

## Compound Symbol

Consider two information sources, *X* & *Y* emitting symbols *x _{i}* ∈

*X*and

*y*∈

_{j}*Y*respectively.

Then, symbol

*c*formed by the combination of symbols from

_{ij}*X*&

*Y*is called

**Compound Symbol**. The compound symbol therefore is an ordered pair ⟨

*x*,

_{i}*y*⟩ taken from a

_{j}**set of symbols**

*C*, given by

## Word, *w*_{i} : A special case of compound symbol *c*_{ij}

_{i}

_{ij}

Consider an information source, *X* emitting *n* symbols, *x _{i}* ∈

*X*into (say) a shift-register. Thus it emits

*n*–symbols in parallel.

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

*w*.

_{i}
The symbol *w _{i}* therefore is an ordered pair of the symbol

*x*in specific positions ⟨

_{i}*x*

_{i}_{,0},

*x*

_{i}_{,1},

*x*

_{i}_{,2}, …,

*x*

_{i}_{,n−1}⟩ taken from a set of all possible

*w*symbols

_{i}*W*is given by

*w*| ≜

_{i}*n*

*W*| ≤ |

*X*|

^{n}

### Illustration of | *w*_{i} | ≜ *n* and | *W* | ≤ | *X* |^{n}

_{i}

Imagine *X* = {*p*, *q*}, thus | *X* | = 2 with possible words; *w _{0}* = ⟨

*p*,

*q*⟩,

*w*= ⟨

_{1}*q*,

*p*⟩,

*w*= ⟨

_{2}*p*,

*p*⟩ and,

*w*= ⟨

_{3}*q*,

*q*⟩.

Therefore, |

*w*| = 2 and |

_{i}*W*| = |

*X*|

^{2}= 2

^{2}= 4.

Notice that if the rule is such that no symbol in a word

*w*can repeat then

_{i}*W*= {

*w*,

_{0}*w*,

_{1}*w*

_{2}*w*

_{3}*W*| = 2. Hence, the upper bound for |

*W*| is 4.

Hence, for the general case

*W*| ≤ |

*X*|

^{2}.