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Graph Theory – Exercise 2.4.5 (a) (i) by Blackburn, de Rijke and Venema's "Modal Logic"

I just want a hint as this is a textbook exercise and I should do it myself.

The problem is something like this:

To let $ C_ ell $ be the cycle of length $ ell + 1 $ (be excruciating
specifically, this is the $ {R } $-Model of an areflexive, symmetrical binary file
relationship $ R $ on the set $ {a_0, dots, a_ ell } $ of $ ell + 1 $ Elements,
$ ell ge 2 $, satisfying $$ a_0Ra_1R cdots a_ ell R 0 $$ and so that
$ a_i Ra_j $ for no other $ i, j $). For an integer $ k $ satisfying $ 2 ^ k
le ell $
, if $ phi $ is a $ R $Sentence with at most quantifier depth
$ k $, show that $$ C_ ell models phi iff C_ ell sqcup C_ ell
models phi. $$

I tried on an induction $ k $, but I didn't get it to work: subtracting quantifiers usually turns a sentence into a non-sentence. My main difficulty is understanding where the "$ 2 ^ k $"comes from. It is true that every induced subgraph of $ C_ ell sqcup C_ ell $ at most on $ ell / 2 $ Corner points are isomorphic to an induced subgraph of $ C_ ell $ (Simply place the connected components through dummy corner points – you will not go out!). Why is that wrong? $ 2k le ell $ Instead of $ 2 ^ k le ell $?

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I am not an owner or administrator. Information was only put up for discussion here.

FORRISE – forrise.pw

IPB Image

Investment plans:
245% AFTER 20 MINUTES, 295% AFTER 40 MINUTES, 435% AFTER 50 MINUTES, 625% AFTER 60 MINUTES, 835% AFTER 80 MINUTES, 1025% AFTER 120 MINUTES

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IPB ImageIPB ImageIPB Image

Paid to payhyip.net $ 24.88
Program description:
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[Vn5socks.net] Auto Update 24/7 – Good Socks 4h30 PM
LIVE ~ 47.94.244.27:7302 | 0.154 | Ottawa | ON | k1y4h7 | Canada | Checked at vn5socks.net
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Rating: 5