Prove that the three basic connectives 'or', 'and' and 'not' can all be written in terms of the single connective 'notand' where 'P notand Q' is interpreted as 'not(P and Q)'.
So I made truth tables for ( P or Q) and (P and Q). I am not sure what the basic connective 'not' is. Is it (not P) / not(Q), or the negation of the individual statements? I figured out that . But I couldnt figure out the forms for or .
Any help is appreciated. Thanks.
not P = not(P and P) = P notand P.
P and Q = (P and Q) or (P and Q)
= not(not(P and Q) and not(P and Q)) (De Morgan's laws)
= not((P notand Q) and (P notand Q)) (Definition of notand)
= (P notand Q) notand (P notand Q). (Definition of notand)
P or Q = not(not P and not Q) (De Morgan's Laws)
= not((P notand P) and (Q notand Q)) (Using not P = P notand P)
= (P notand P) notand (Q notand Q). (Definition of notand)
Is there something simpler?
This function has as interesting history. If is known as Sheffer’s stroke function after H.M.Sheffer who named it. Although, CS Peirce actually discovered it and its properties and Quine called it the alternate denial function..
It is most often defined as in the truth-table below.
I think that using the truth-table is easy to see what JakeD found to be true.