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Tableau for ¬((p∧q)∨r)

Draw the tableau for the formula Φ = ¬((p∧q)∨r), where ¬ denotes the negation of a variable, ∨ denotes the disjunction of variables and ∧ denotes the conjunction of variables.

Formula Φ = ¬((p ∧ q) ∨ r).

Rules. ¬(A ∨ B) is an α-formula and gives ¬A and ¬B on the same branch. ¬(p ∧ q) is a β-formula and splits into ¬p | ¬q.

Tableau

          ¬((p ∧ q) ∨ r)
                |
           ¬(p ∧ q)
                |
               ¬r
              /    \
            ¬p      ¬q
             o       o
BranchLiteralsStatus
Left¬r, ¬popen
Right¬r, ¬qopen

Both branches are open, so Φ is satisfiable. Models: r = F with p = F, or r = F with q = F. (Φ is false whenever r is true, or when p and q are both true.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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