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What do you mean by the term strict part of a binary relation?

How can it be used to define minimal element for any set with relation?

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Please provide more context. – Henning Makholm Oct 8 '11 at 17:32
@user774025: Maybe it means take out all pairs of the form $(x,x)$? – Weltschmerz Oct 8 '11 at 17:42
Do you mean:"the less strict definition" of the binary relation? – pedja Oct 8 '11 at 17:44
@user774025,… – pedja Oct 8 '11 at 18:03
No, not the less "the less strict definition" of the binary relation. Given a binary relation, you can obtain obtain another binary relation called its strict part by removing some elements from the old binary relation. But I don't exactly which elements are removed. – Mohan Oct 8 '11 at 18:10
up vote 2 down vote accepted

It's hard to say without more context, but it seems like you are to take out pairs of the form $(x,x)$ from your relation (i.e. dropping the condition of reflexivity). This is similar to how "less than or equal to" gives rise to "strictly less than".

Given a strict relation, you can find a minimal element of a finite set by taking a descending chain $x_1 > x_2 > \cdots > x_n$. Since the set is finite, the chain will indeed terminate at a minimal element.

If the set is infinite, there may not be a minimal element under every relation. For example, in the real numbers with usual "less than" relation there can be descending chains with no minimal element.

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I haven't heard the term before, but this is what I would immediately assume it means, for precisely the reason given here. – Michael Hardy Oct 9 '11 at 1:01

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