A set $S$ together with partial order $\ll$ is called a well order if it has no infinite descending chains, i.e. there is no infinite sequence $x_1, x_2,\ldots$ of elements from $S$ such that $x_{i+1} \ll x_i$ and $x_{i+1} \neq x_i$ for all $i$.
Consider the set of all words (finite sequence of letters $a - z$), denoted by $W$, in dictionary order.
I understood how it is a partial order. But I did not understand why it is a well order? According to the definition given in the question, a partial order ≪ is called a well order if it has no infinite descending chains.
But if we pick random word like ab, it has infinite descendants like aa, aaa, aaaaaa, .....
How is this a well order?
Answer -> E)well order
Minimal Element is 'a', it is less than all elements !
a) False, after aa, we can have ab. Then aba,abb,abc.. Not limited to 24
b) False. after aa, we can have ab,aba,abc.. In fact ab(a-z)*. Not limited to 2^{24}
C)False. Why not partial order ? Dictionary order is partial order ! It is Reflexive, Antysymmetric & Transitive. Even defination of wikipedia says it is !
D) False.Dictionary order is well order .
Defination of Dictionary order -> Ref -> https://en.wikipedia.org/wiki/Lexicographical_order
Given two partially ordered sets A and B, the lexicographical order on the Cartesian product A × B is defined as
Is it possible to know how many words are there from aa to az?
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