Using interpretations

Show that each of the following is neither a validity nor a contradiction:

1. $D(a)\wedge D(b)$

2. $\exists x\,T(x,h)$

3. $P(m)\wedge\neg\forall x\,P(x)$

4. $\forall z\,J(z)\leftrightarrow\exists y\,J(y)$

5. $\forall x(W(x,m,n)\vee\exists y\,L(x,y))$

6. $\exists x(G(x)\rightarrow\forall y\,M(y))$

7. $\exists x(x=h\wedge x=i)$

Show that the following pairs of sentences are not logically equivalent.

1. $J(a)$, $K(a)$

2. $\exists x\,J(x)$, $J(m)$

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

4. $\exists x\,P(x)\rightarrow Q(c)$, $\exists x(P(x)\rightarrow Q(c))$

5. $\forall x(P(x)\rightarrow\neg Q(x))$, $\exists x(P(x)\wedge\neg Q(x))$

6. $\exists x(P(x)\wedge Q(x))$, $\exists x(P(x)\rightarrow Q(x))$

7. $\forall x(P(x)\rightarrow Q(x))$, $\forall x(P(x)\wedge Q(x))$

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

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

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