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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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.