Proof of functional completeness and adequate of logical operators

Asked Oct 29 '20 at 18:27

Active Nov 17 '20 at 04:00

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a) Prove that {∨, ∧, ¬} is functional complete

b) Prove that the singleton sets {|} and {↓} where

$$ \begin{array}{cc|ccccc|c} x & y & x \mid y & & x & y & x \downarrow y \\ \hline 0 & 0 & 1 & & 0 & 0 & 1 \\ 0 & 1 & 1 & & 0 & 1 & 0 \\ 1 & 0 & 1 & & 1 & 0 & 0 \\ 1 & 1 & 0 & & 1 & 1 & 0 \end{array} $$

are functional complete. The connective | is called Sheffer stroke and the connective ↓ is called Pierce arrow.

Table image

c) Prove that the sets {∨, ∧}, {→, ∨}, {→, ∧} are not adequate sets of connectives.

edited Nov 17 '20 at 04:00

Wolgwang

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asked Oct 29 '20 at 18:27

carciofinospinoso

And many similar posts visible on the right of the screen... – Mauro ALLEGRANZA Oct 29 '20 at 18:31
but it doesn't resolve my question.. – carciofinospinoso Oct 29 '20 at 18:36
Do you know how to prove functional completeness ? – Mauro ALLEGRANZA Oct 29 '20 at 18:59
In the linked post there is the proof that ${ \lor, \lnot }$ is functional complete. This means that $\land$ can be expressed with them. Thus, also ${ \land, \lor, \lnot }$ is complete. – Mauro ALLEGRANZA Oct 29 '20 at 19:01
For Sheffer and Peirce connectives, repeat the procedure, showing that both are able to express $\land, \lor$ and $\lnot$ (you van see Wiki's entries). – Mauro ALLEGRANZA Oct 29 '20 at 19:02

Proof of functional completeness and adequate of logical operators

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