Succeed is Dual to Precede

Theorem

Let $\left({S, \preceq}\right)$ be an ordered set.

Let $a, b \in S$.

The following are dual statements:

$a$ succeeds $b$

$a$ precedes $b$

Proof

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

$b \preceq a$

The dual of this statement is:

$a \preceq b$

by Dual Pairs (Order Theory).

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

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

$\blacksquare$

Also see

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