Chapter 1 - Section 1.4 - Predicates and Quantifiers - Exercises - Page 54: 26

a) $\exists x P(x)$ $\exists x (A(x) \land P(x))$ $\exists x (A(x) \land P(x,Uzbekistan))$. b) $\forall x (Q(x) \land R(x))$ $\forall x [B(x) \rightarrow (Q(x) \land R(x)]$ $\forall x [B(x) \rightarrow (Q(x,calculus) \land Q(x, C++)]$ c) $\neg \exists x (S(x) \land T(x))$ $\neg \exists x (A(x) \land S(x) \land T(x))$ $\neg \exists x (A(x) \land S(x,bicycle) \land S(x,motorcycle))$ d) $\exists x (\neg U(x))$ $\exists x (A(x) \land\neg U(x))$ $\exists x (A(x) \land\neg U(x,happy))$ e) $\forall x V(x)$ $\forall x (A(x) \rightarrow V(x))$ $\forall x (A(x) \rightarrow V(x,Twentieth))$

a) Let the first domain be the people in your school and the second domain all the people in the world. Let A(x) mean ' x is in your school” and P(x) means "x has visited Uzbekistan" and P(x.y) means "x has visited y". $\exists x P(x)$ $\exists x (A(x) \land P(x))$ $\exists x (A(x) \land P(x,Uzbekistan))$. b)Let the first domain be the people in your class and the second domain all the people in the world. Let B(x) mean "x is in your class" and Q(x) means "x has studied calculus" and R(x) means "x has studied C++". Q(x.y) means "x has studied y". $\forall x (Q(x) \land R(x))$ $\forall x [B(x) \rightarrow (Q(x) \land R(x)]$ $\forall x [B(x) \rightarrow (Q(x,calculus) \land Q(x, C++)]$ c) Let the first domain be the people in your school and the second domain all the people in the world. Let A(x) mean "x is in your school" and S(x) means "x owns a bicycle" and T(x) means " x owns a motorcycle S(x,y) means "x owns a y“. $\neg \exists x (S(x) \land T(x))$ $\neg \exists x (A(x) \land S(x) \land T(x))$ $\neg \exists x (A(x) \land S(x,bicycle) \land S(x,motorcycle))$ d) Let the first domain be the people in your school and the second domain all the people in the world. Let A(x) mean is in your school” and U(x) means "x is happy”. U(x.y) means "x is y". $\exists x (\neg U(x))$ $\exists x (A(x) \land\neg U(x))$ $\exists x (A(x) \land\neg U(x,happy))$ e) Lot the first domain be the people in your school and the second domain all the people in the world. Let A(x) mean x is in your school " and V{x) means ”x was born in the twentieth century", V(x, y) means "x was born in the y century”. $\forall x V(x)$ $\forall x (A(x) \rightarrow V(x))$ $\forall x (A(x) \rightarrow V(x,Twentieth))$