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

Ebone6v

Answered question

2022-08-13

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.

Answer & Explanation

Gaige Burton

Gaige Burton

Beginner2022-08-14Added 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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