Functionally Complete Logical Connectives/NAND

Theorem

The singleton set containing the following logical connective:

$\set \uparrow$: NAND

is functionally complete.


Proof

From Functionally Complete Logical Connectives: Negation and Conjunction, any boolean expression can be expressed in terms of $\land$ and $\neg$.


From NAND with Equal Arguments:

$\neg p \dashv \vdash p \uparrow p$


From Conjunction in terms of NAND:

$p \land q \dashv \vdash \paren {p \uparrow q} \uparrow \paren {p \uparrow q}$

demonstrating that $p \land q$ is expressed solely in terms of $\uparrow$.


Thus any boolean expression can be represented solely in terms of $\uparrow$.

That is, $\set \uparrow$ is functionally complete.

$\blacksquare$


Also see


Sources

  • 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants
  • 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $2$: The Propositional Calculus $2$: $3$ Truth-Tables: Exercise $2 \ \text{(ii)}$
  • 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 1$: Some mathematical language: Connectives
  • 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.4.2$
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