Home / Computer science / 3-satisfiability (3SAT) / Explore / Case 4: More than three literals / Proof / Case 1: \(c\) is satisfiable
Case 1B: \(\ell_{k - 1} = 1\) or \(\ell_k = 1\)
Setting
\begin{equation*} y_i = 1 \quad \forall i \end{equation*}satisfies all clauses:
\begin{align*} c'_1 & = (\ell_1 \lor \ell_2 \lor y_1) \\ & = (\ell_1 \lor \ell_2 \lor \boxed{1}) \\ & = 1 \\[2ex] c'_2 & = (\lnot y_1 \lor \ell_3 \lor y_2) \\ & = (\lnot 1 \lor \ell_3 \lor \boxed{1}) \\ & = 1 \\[1ex] \vdots & \\[2ex] c'_{k - 1} & = (\lnot y_{k - 4} \lor \ell_{k - 2} \lor y_{k - 3}) \\ & = (\lnot 1 \lor \ell_{k - 2} \lor \boxed{1}) \\ & = 1 \\[2ex] c'_{k - 2} & = (\lnot y_{k - 3} \lor \ell_{k - 1} \lor \ell_k) \\ & = (\lnot 1 \lor \boxed{\ell_{k - 1}} \lor \boxed{\ell_k}) \\ & = 1. \end{align*}