According to wikipedia, the set {^, ¬} is functionally complete. But is there any 2-set functionally complete set with XOR (e.g. (¬A) ⊕ A is always true).

I'm looking for a 2-set functionally complete with xor e.g. is {&,⊕} or {¬.⊕} functionally complete?

  • 2
    $\begingroup$ See wikipedia: "Emil Post proved that a set of logical connectives is functionally complete if and only if it is not a subset of any of the following sets of connectives ... " en.wikipedia.org/wiki/Functional_completeness $\endgroup$ – Myself Mar 17 '13 at 19:28

$\{{\neg},{\oplus}\}$ is not complete; see this question.

$\{{\land},{\oplus}\}$ is not complete either, because if all the inputs are false, then the output is always false too.

However $\{{\to}, {\oplus}\}$ is complete, because $\{{\neg},{\to}\}$ is known to be, and $(A\to A)\oplus A$ is equivalent to $\neg A$.

  • $\begingroup$ Perfect. I'm glad you understood. $\endgroup$ – Niklas Rosencrantz Mar 17 '13 at 19:36

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.