axametnica
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- Apr 9, 2020
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hello, i want to make sure if my steps are right. My homework is: convert to conjunctive normal form.
∀x(p(x) ⇒ ∃y(q(x, y) ∧ q(z) ∨ ¬r(y, z)))
My steps are:
1.) ∀x(¬p(x) ∨ ∃y(q(x, y) ∧ q(z) ∨ ¬r(y, z)))
2.) ∀x∃y(¬p(x) ∨ (q(x, y) ∧ q(z) ∨ ¬r(y, z)))
3.) ∀x(¬p(x) ∨ (q(x, f (x)) ∧ q(z) ∨ ¬r(f (x), z)))
4.) ¬p(x) ∨ (q(x, f (x)) ∧ q(z) ∨ ¬r(f (x), z))
5.) ¬p(x) ∨ (¬r(f (x), z) ∨ (q(x, f (x)) ∧ q(z)))
6.) ¬p(x) ∨ ((¬r(f (x), z) ∨ q(x, f (x))) ∧ (¬r(f (x), z) ∨ q(z)))
7.) (¬p(x) ∨ (¬r(f (x), z) ∨ q(x, f (x)))) ∧ (¬p(x) ∨ (¬r(f (x), z) ∨ q(z)))
CNF form: (¬r(f (x), z) ∨ q(x, f (x)) ∨ ¬p(x)) ∧ (¬r(f (x), z) ∨ q(z) ∨ ¬p(x))
Is it correct please? Thank you so much for all answer
∀x(p(x) ⇒ ∃y(q(x, y) ∧ q(z) ∨ ¬r(y, z)))
My steps are:
1.) ∀x(¬p(x) ∨ ∃y(q(x, y) ∧ q(z) ∨ ¬r(y, z)))
2.) ∀x∃y(¬p(x) ∨ (q(x, y) ∧ q(z) ∨ ¬r(y, z)))
3.) ∀x(¬p(x) ∨ (q(x, f (x)) ∧ q(z) ∨ ¬r(f (x), z)))
4.) ¬p(x) ∨ (q(x, f (x)) ∧ q(z) ∨ ¬r(f (x), z))
5.) ¬p(x) ∨ (¬r(f (x), z) ∨ (q(x, f (x)) ∧ q(z)))
6.) ¬p(x) ∨ ((¬r(f (x), z) ∨ q(x, f (x))) ∧ (¬r(f (x), z) ∨ q(z)))
7.) (¬p(x) ∨ (¬r(f (x), z) ∨ q(x, f (x)))) ∧ (¬p(x) ∨ (¬r(f (x), z) ∨ q(z)))
CNF form: (¬r(f (x), z) ∨ q(x, f (x)) ∨ ¬p(x)) ∧ (¬r(f (x), z) ∨ q(z) ∨ ¬p(x))
Is it correct please? Thank you so much for all answer