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Topic: Transitive closure that is reflexive?
Replies: 4   Last Post: Sep 24, 2001 1:14 AM

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Dave Seaman

Posts: 2,446
Registered: 12/6/04
Re: Transitive closure that is reflexive?
Posted: Sep 24, 2001 1:14 AM
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In article <9ok0rg$sog$>, <> wrote:

>I am kinda new to discrete mathematics. I am reading a book by Harry
>Lewis, in which I find some confusing sentences. Here is one:

>a binary relation that is not reflexive but has a transitive closure that
>is reflexive.

That is not a sentence. It has no verb. Are you saying there is some
particular binary relation that has these properties? It certainly is
not true of all binary relations.

>I understand 'reflexive', but just don't get 'transitive closure', and a
>'transitive closure that is reflexive'. So confusing.

>Can you guys help explain this to me with an example? Thanks a lot.

Consider the not-equal relation R on any set X having at least two
members. R is not reflexive, but its transitive closure R' is reflexive,
because for each x in X there exists y in X such that xRy and yRx
(meaning x != y and y != x). Hence (x,x) is in R', the transitive
closure of R.

Dave Seaman
Amnesty International calls for new trial for Mumia Abu-Jamal

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