GATE CSE
First time here? Checkout the FAQ!
x
+2 votes
217 views

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.

  1. Between $``aa"$ and $``az"$ there are only $24$ words.
  2. Between $``aa"$ and $``az"$ there are only $2^{24}$ words.
  3. $W$ is not a partial order.
  4. $W$ is a partial order but not a well order.
  5. $W$ is a well order.
asked in Set Theory & Algebra by Veteran (29.5k points)   | 217 views

2 Answers

+2 votes

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 224

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

(a,b) ≤ (a′,b′) if and only if a < a′ or (a = a′ and bb′).
The result is a partial order. If A and B are each totally ordered, then the result is a total order as well. The lexicographical order of two totally ordered sets is thus a linear extension of their product order.
 
 
 

 

answered by Veteran (41.5k points)  
Isn't the sequence $``z" \;\gg\; ``yz" \;\gg\; ``yyz" \;\gg\; ``yyyz" \cdots$ a sequence such that $x_{i+1} \ll x_i$ and $x_{i+1} \neq x_i$?
0 votes
Ans will be (c)W is not partial order
answered by Veteran (53.7k points)  
why?
Because W={aa,ab...........az,ba,bb,............bz,.............za,zb,......zz}

all are possible

if we allow {ab,ba} then it will be a symmetric relation . So partial order is not possible
The order is "dictionary" and it is specified in question. So, ab << ba.
ok then dictionary order means ba,ca,cb those are not possible right?

and well order set means chains like 1-----2----------4-------12, right?

plz ans here
why "ba", "ca" etc not possible? Any word is possible in a dictionary.

aaaaaa << ba

So, options A and B are false.

I'm not sure of well order- but answer here is well order.
yes it is woset because it is toset and lowest element is fixed - 'a'.

Related questions



Top Users May 2017
  1. akash.dinkar12

    3140 Points

  2. pawan kumarln

    1606 Points

  3. sh!va

    1580 Points

  4. Arjun

    1316 Points

  5. Devshree Dubey

    1230 Points

  6. Angkit

    1020 Points

  7. Debashish Deka

    1012 Points

  8. Bikram

    970 Points

  9. LeenSharma

    796 Points

  10. srestha

    658 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 22 - 28
  1. pawan kumarln

    232 Points

  2. jjayantamahata

    106 Points

  3. joshi_nitish

    106 Points

  4. Ahwan

    96 Points

  5. Aditya GN

    63 Points


22,717 questions
29,045 answers
65,029 comments
27,454 users