Collection of Maths Problems

Existence of maximal and largest elements

Task number: 4152

• Variant

  Each non-empty finite partial order has a maximum element.

• Solution

  By induction according to the size of the base set \(X\).

  If it is single-element, then this element is maximal.

  Choose any \( a \in X \). From the inductive hypothesis, \( (X \setminus a, \leq) \) has the maximum element \( b \). If \( a > b \), then \( a \) is the maximum element on \( (X, \leq) \), otherwise \( b \) remains the maximum element even on \( (X, \leq) \).

• Variant

  Each non-empty finite linear order has the largest element.

• Solution

  By induction according to the size of the base set \( X \).

  If it is single-element, then this element is the largest.

  Choose any \( a \in X \). From the inductive hypothesis, \( (X \setminus a, \leq) \) has the largest element \( b \). If \( a > b \), then \( a \) is the largest element on \( (X, \leq) \) from the transitivity, otherwise \( b \) remains the largest element even on \( (X, \leq) \), because \( a \) cannot be incomparable with \( b \) due to the linearity of the order.