12,089
edits
Changes
cleanup
'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd.
'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits is odd.
A + B = (A ∧ !B) ∨ (!A ∧ B)
= {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
= {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
= (!A ∨ !B) ∧ (A ∨ B)
= !(A ∧ B) ∧ (A ∨ B)
p + q = (p ∧ !q) ∨ (!p ∧ B)
= {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
= {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
= (!p ∨ !q) ∧ (p ∨ q)
= !(p ∧ q) ∧ (p ∨ q)
p + q = (p ∧ ~q) ∨ (~p ∧ q)
= ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
= ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
= (~p ∨ ~q) ∧ (p ∨ q)
= ~(p ∧ q) ∧ (p ∨ q)
: <math>\begin{matrix}
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
& = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\
& = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\
& = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\
& = & \lnot (p \land q) & \land & (p \lor q)
\end{matrix}</math>
====[[Logical implication]]====
====[[Logical implication]]====
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|}
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==Relational Tables==
==Relational Tables==