Suppose R is a partial order on a set A, and B sube A. Prove that, if R is a total order and b is a minimal element of B, then b is the smallest element of B.

Ebone6v 2022-08-13 Answered
Suppose R is a partial order on a set A, and B A.
Prove that, if R is a total order and b is a minimal element of B, then b is the smallest element of B.
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Answers (1)

Gaige Burton
Answered 2022-08-14 Author has 16 answers
It's easy. Let denote the total order. Let b be any minimal element of B, and u another element of B different from b. Then u b or b u. If u b then you get u = b, a contradiction. Thus, b u, so b is minimum.
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