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By Mark V. Lawson

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Thus i=3 j=3 xor(cij1 , cij2 , cij3 ). B= i=1 j=1 Then B is true precisely when each cell of the grid contains exactly one of the numbers 1, 2, 3. • In each row, each of the numbers 1, 2, 3 must occur exactly once. For each 1 ≤ i ≤ 3, define Ri = xor(ci11 , ci21 , ci31 ) ∧ xor(ci12 , ci22 , ci32 ) ∧ xor(ci13 , ci23 , ci33 ). Then Ri is true when each of the numbers 1, 2, 3 occurs exactly once in the cells in row i. Define R = i=3 i=1 Ri . • In each column, each of the numbers 1, 2, 3 must occur exactly once.

7 Normal forms A normal form is a particular way of writing a wff. We begin with a normal form that is a stepping stone to two others that are more important. 7. NORMAL FORMS 35 A wff is in negation normal form (NNF) if it is constructed using only ∧, ∨ and literals. Recall that a literal is either an atom or the negation of an atom. 1. Every wff is logically equivalent to a wff in NNF. Proof. Let A be a wff in PL. First, replace any occurrences of x ⊕ y by ¬(x ↔ y). Second, replace any occurrences of x ↔ y by (x → y) ∧ (y → x).

An B is not a valid argument. Then some assignment of truth values to the atoms makes A1 , . . , An true and B false. This means that A1 ∧ . . ∧ An is true and B is false. But this contradicts that we are given that A1 ∧ . . ∧ An B is a valid argument. 2. We prove that A B precisely when A → B. Suppose that A B is a valid argument and that A → B is not a tautology. Then there is some assignment of the truth values to the atoms that makes A true and B false. But this contradicts that A B is a valid argument.