Exercise C

Show that the following sentences are jointly satisfiable:

1. $M(a),\neg N(a),Pa,\neg Q(a)$

2. $L(e,e),L(e,g),\neg L(g,e),\neg L(g,g)$

3. $\neg(M(a)\wedge\exists x\,A(x)),Ma\vee F(a),\forall x(F(x)\rightarrow A(x))$

4. $M(a)\vee M(b),M(a)\rightarrow\forall x\neg M(x)$

5. $\forall y\,G(y),\forall x(G(x)\rightarrow H(x)),\exists y\neg I(y)$

6. $\exists x(B(x)\vee A(x)),\forall x\neg C(x),\forall x\bigl[(A(x)\wedge B(x))\rightarrow C(x)\bigr]$

7. $\exists x\,X(x),\exists x\,Y(x),\forall x(X(x)\leftrightarrow\neg Y(x))$

8. $\forall x(P(x)\vee Q(x)),\exists x\neg(Q(x)\wedge P(x))$

9. $\exists z(N(z)\wedge O(z,z)),\forall x\forall y(O(x,y)\rightarrow O(y,x))$

10. $\neg\exists x\forall y\,R(x,y),\forall x\exists y\,R(x,y)$

11. $\neg R(a,a)$, $\forall x(x=a\vee R(x,a))$

12. $\forall x\forall y\forall z[(x=y\vee y=z)\vee x=z]$, $\exists x\exists y\ \neg x=y$

13. $\exists x\exists y((Z(x)\wedge Z(y))\wedge x=y)$, $\neg Z(d)$, $d=e$