Exercise C

Following the diagram conventions introduced at the end of §23, consider the following interpretation:

(atom1) at (0,2) 1; (atom2) at (2,2) 2; (atom4) at (0,0) 3; (atom5) at (2,0) 4; (atom6) at (4,0) 5; [-¿, thick] (atom1)+(-0.15,0.15) arc (-330:-30:.3); [-¿, thick] (atom6)+(0.15,-0.15) arc (-150:150:.3); [-¿, thick] (atom1) – (atom2); [-¿, thick] (atom1) – (atom4); [¡-¿, thick] (atom1) – (atom5); [-¿, thick] (atom1) – (atom6); [-¿, thick] (atom2) – (atom6);

Determine whether each of the following sentences is true or false in that interpretation:

1. $\exists x\,R(x,x)$

2. $\forall x\,R(x,x)$

3. $\exists x\forall y\,R(x,y)$

4. $\exists x\forall y\,R(y,x)$

5. $\forall x\forall y\forall z((R(x,y)\wedge R(y,z))\rightarrow R(x,z))$

6. $\forall x\forall y\forall z((R(x,y)\wedge R(x,z))\rightarrow R(y,z))$

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

8. $\forall x(\exists y\,R(x,y)\rightarrow\exists y\,R(y,x))$

9. $\exists x\exists y(\neg x=y\wedge R(x,y)\wedge R(y,x))$

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

11. $\exists x\forall y(R(y,x)\leftrightarrow x=y)$

12. $\exists x\exists y(\neg x=y\wedge R(x,y)\wedge\forall z(R(z,x)\leftrightarrow y=z))$