Strictly Succeed is Dual to Strictly Precede

Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $a, b \in S$.

The following are dual statements:

$a$ strictly succeeds $b$

$a$ strictly precedes $b$

Proof

By definition, $a$ strictly succeeds $b$ if and only if:

$b \preceq a$ and $b \ne a$

The dual of this statement is:

$a \succeq b$ and $b \ne a$

by Dual Pairs (Order Theory).

By definition, this means $a$ strictly precedes $b$.

The converse follows from Dual of Dual Statement (Order Theory).

$\blacksquare$

Also see

• Duality Principle (Order Theory)
• Succeed is Dual to Precede