Functionally Complete Logical Connectives/Negation and Conjunction

From ProofWiki
Jump to navigation Jump to search

Theorem

The set of logical connectives:

$\set {\neg, \land}$: Not and And

is functionally complete.


This article is complete as far as it goes, but it could do with expansion.
In particular: to adopt the notion of Independent Set of Statements
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Proof

From Functionally Complete Logical Connectives: Negation, Conjunction, Disjunction and Implication, all sixteen of the binary truth functions can be expressed in terms of $\neg, \land, \lor, \implies$.


From Conjunction and Implication, we have that:

$p \implies q \dashv \vdash \neg \paren {p \land \neg q}$

From De Morgan's laws: Disjunction, we have that:

$p \lor q \dashv \vdash \neg \paren {\neg p \land \neg q}$

So any instance of either $\implies$ or $\lor$ can be replaced identically with one using just $\neg$ and $\land$.

It follows that $\set {\neg, \land}$ is functionally complete.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Functionally_Complete_Logical_Connectives/Negation_and_Conjunction&oldid=587992"