Given boolean variables x1, x2, and x3, construct a propositional formula φ that
holds if and only if an odd number of the variables is true.
DNF: (x3 ∧ x2 ∧ x1) ∨ (x3 ∧ ¬x2 ∧ ¬x1) ∨ (¬x3 ∧ x2 ∧ ¬x1) ∨ (¬x3 ∧ ¬x2 ∧ x1)
In another Question:
Given boolean variables x1, x2, x3, and x4, construct a propositional formula φ
that holds if and only if an even number of the variables is true.
DNF:
(x4 ∧ x3 ∧ x2 ∧ x1) ∨ (x4 ∧ x3 ∧ ¬x2 ∧ ¬x1) ∨
(x4 ∧ ¬x3 ∧ x2 ∧ ¬x1) ∨ (x4 ∧ ¬x3 ∧ ¬x2 ∧ x1) ∨ (¬x4 ∧ x3 ∧ x2 ∧ ¬x1) ∨ (¬x4 ∧ x3 ∧ ¬x2 ∧ x1) ∨ (¬x4 ∧ ¬x3 ∧ x2 ∧ x1) ∨ (¬x4 ∧ ¬x3 ∧ ¬x2 ∧ ¬x1)
Why we are taking this (¬x4 ∧ ¬x3 ∧ ¬x2 ∧ ¬x1)
As far as I understand if x1=x2=x3=x4=0 it should be false?
Correct me please.
Reference papers 2015