Skip to content
January 25, 2012 / porton

Some new theorems

I added the following theorems to Funcoids and Reloids article. The theorems are simple to prove but are surprising, as do something similar to inverting a binary relation which is generally neither monovalued nor injective.

Proposition Let f, g, h are binary relations. Then g \circ f \not\asymp h \Leftrightarrow g \not\asymp h \circ f^{- 1}.

Theorem Let A, B, C are sets, f \in \mathsf{FCD} (A ; B), g \in \mathsf{FCD} (B ; C), h \in \mathsf{FCD}(A ; C). Then

g \circ f \not\asymp h \Leftrightarrow g \not\asymp h \circ f^{- 1} .

Theorem Let A, B, C are sets, f \in \mathsf{RLD} (A ; B), g \in \mathsf{RLD} (B ; C), h \in \mathsf{RLD}(A ; C). Then

g \circ f \not\asymp h \Leftrightarrow g \not\asymp h \circ f^{- 1} .

The above theorems are the key for describing product funcoids, a task I previously got stuck. Now I can continue my research.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: