Modal logic

Problem 2. Provide counterexample that prove the following failures of entailment, and explain why they are counterexamples. Search for these counterexamples by using the tableaux method: 1. O(A VB)* OAV OB 2. K OA DOA 3. 4. O(AVC), A V (BVC)O(A VB VC Negation: (1) (2) --A, X A, X Conditional: (1) A B, x (1) (2) (3) -(A + B), x A, X -B, (2) -A, B,x Disjunction: (1) AVB, X (1) (2) (3) -(A VB), x -A, X -B, X (2) A, X B, X Conjunction: (1) -(A&B), X (1) (2) (3) A&B, x A, X B, X (2) -A, x B x Biconditional: (1) AB, X (1) -(A + B), X (2) (3) A, X A,x (2) B, X -B, X (3) A, XA, X -B, X B, X Necessity: (1) (2) (3) DA, X XRy (1) (2) -DA, X -A, X A, Y A, X : xRy (1) (2) Possibility: (1) (2) (3) (4) LOA, X O-A, X y must be new A, Y Problem 2. Provide counterexample that prove the following failures of entailment, and explain why they are counterexamples. Search for these counterexamples by using the tableaux method: 1. O(A VB)* OAV OB 2. K OA DOA 3. 4. O(AVC), A V (BVC)O(A VB VC Negation: (1) (2) --A, X A, X Conditional: (1) A B, x (1) (2) (3) -(A + B), x A, X -B, (2) -A, B,x Disjunction: (1) AVB, X (1) (2) (3) -(A VB), x -A, X -B, X (2) A, X B, X Conjunction: (1) -(A&B), X (1) (2) (3) A&B, x A, X B, X (2) -A, x B x Biconditional: (1) AB, X (1) -(A + B), X (2) (3) A, X A,x (2) B, X -B, X (3) A, XA, X -B, X B, X Necessity: (1) (2) (3) DA, X XRy (1) (2) -DA, X -A, X A, Y A, X : xRy (1) (2) Possibility: (1) (2) (3) (4) LOA, X O-A, X y must be new A, Y