### Artificial IntelligenceAIMA Exercises

In each of the following we give an English sentence and a number of candidate logical expressions. For each of the logical expressions, state whether it (1) correctly expresses the English sentence; (2) is syntactically invalid and therefore meaningless; or (3) is syntactically valid but does not express the meaning of the English sentence.
1. Every cat loves its mother or father.
1. ${\forall\,x\;\;} {Cat}(x) {\:\;{\Rightarrow}\:\;}{Loves}(x,{Mother}(x)\lor {Father}(x))$.
2. ${\forall\,x\;\;} \lnot {Cat}(x) \lor {Loves}(x,{Mother}(x)) \lor {Loves}(x,{Father}(x))$.
3. ${\forall\,x\;\;} {Cat}(x) \land ({Loves}(x,{Mother}(x))\lor {Loves}(x,{Father}(x)))$.
2. Every dog who loves one of its brothers is happy.
1. ${\forall\,x\;\;} {Dog}(x) \land (\exists y\ {Brother}(y,x) \land {Loves}(x,y)) {\:\;{\Rightarrow}\:\;}{Happy}(x)$.
2. ${\forall\,x,y\;\;} {Dog}(x) \land {Brother}(y,x) \land {Loves}(x,y) {\:\;{\Rightarrow}\:\;}{Happy}(x)$.
3. ${\forall\,x\;\;} {Dog}(x) \land [{\forall\,y\;\;} {Brother}(y,x) {\;\;{\Leftrightarrow}\;\;}{Loves}(x,y)] {\:\;{\Rightarrow}\:\;}{Happy}(x)$.
3. No dog bites a child of its owner.
1. ${\forall\,x\;\;} {Dog}(x) {\:\;{\Rightarrow}\:\;}\lnot {Bites}(x,{Child}({Owner}(x)))$.
2. $\lnot {\exists\,x,y\;\;} {Dog}(x) \land {Child}(y,{Owner}(x)) \land {Bites}(x,y)$.
3. ${\forall\,x\;\;} {Dog}(x) {\:\;{\Rightarrow}\:\;}({\forall\,y\;\;} {Child}(y,{Owner}(x)) {\:\;{\Rightarrow}\:\;}\lnot {Bites}(x,y))$.
4. $\lnot {\exists\,x\;\;} {Dog}(x) {\:\;{\Rightarrow}\:\;}({\exists\,y\;\;} {Child}(y,{Owner}(x)) \land {Bites}(x,y))$.
4. Everyone’s zip code within a state has the same first digit.
1. ${\forall\,x,s,z_1\;\;} [{State}(s) \land {LivesIn}(x,s) \land {Zip}(x)z_1] {\:\;{\Rightarrow}\:\;}{}$\ $[{\forall\,y,z_2\;\;} {LivesIn}(y,s) \land {Zip}(y)z_2 {\:\;{\Rightarrow}\:\;}{Digit}(1,z_1) {Digit}(1,z_2) ]$.
2. ${\forall\,x,s\;\;} [{State}(s) \land {LivesIn}(x,s) \land {\exists\,z_1\;\;} {Zip}(x)z_1] {\:\;{\Rightarrow}\:\;}{}$\ $[{\forall\,y,z_2\;\;} {LivesIn}(y,s) \land {Zip}(y)z_2 \land {Digit}(1,z_1) {Digit}(1,z_2) ]$.
3. ${\forall\,x,y,s\;\;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {\:\;{\Rightarrow}\:\;}{Digit}(1,{Zip}(x){Zip}(y))$.
4. ${\forall\,x,y,s\;\;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {\:\;{\Rightarrow}\:\;}{}$\ ${Digit}(1,{Zip}(x)) {Digit}(1,{Zip}(y))$.

In each of the following we give an English sentence and a number of candidate logical expressions. For each of the logical expressions, state whether it (1) correctly expresses the English sentence; (2) is syntactically invalid and therefore meaningless; or (3) is syntactically valid but does not express the meaning of the English sentence.
1. Every cat loves its mother or father.
1. ${\forall\,x\;\;} {Cat}(x) {\:\;{\Rightarrow}\:\;}{Loves}(x,{Mother}(x)\lor {Father}(x))$.
2. ${\forall\,x\;\;} \lnot {Cat}(x) \lor {Loves}(x,{Mother}(x)) \lor {Loves}(x,{Father}(x))$.
3. ${\forall\,x\;\;} {Cat}(x) \land ({Loves}(x,{Mother}(x))\lor {Loves}(x,{Father}(x)))$.
2. Every dog who loves one of its brothers is happy.
1. ${\forall\,x\;\;} {Dog}(x) \land (\exists y\ {Brother}(y,x) \land {Loves}(x,y)) {\:\;{\Rightarrow}\:\;}{Happy}(x)$.
2. ${\forall\,x,y\;\;} {Dog}(x) \land {Brother}(y,x) \land {Loves}(x,y) {\:\;{\Rightarrow}\:\;}{Happy}(x)$.
3. ${\forall\,x\;\;} {Dog}(x) \land [{\forall\,y\;\;} {Brother}(y,x) {\;\;{\Leftrightarrow}\;\;}{Loves}(x,y)] {\:\;{\Rightarrow}\:\;}{Happy}(x)$.
3. No dog bites a child of its owner.
1. ${\forall\,x\;\;} {Dog}(x) {\:\;{\Rightarrow}\:\;}\lnot {Bites}(x,{Child}({Owner}(x)))$.
2. $\lnot {\exists\,x,y\;\;} {Dog}(x) \land {Child}(y,{Owner}(x)) \land {Bites}(x,y)$.
3. ${\forall\,x\;\;} {Dog}(x) {\:\;{\Rightarrow}\:\;}({\forall\,y\;\;} {Child}(y,{Owner}(x)) {\:\;{\Rightarrow}\:\;}\lnot {Bites}(x,y))$.
4. $\lnot {\exists\,x\;\;} {Dog}(x) {\:\;{\Rightarrow}\:\;}({\exists\,y\;\;} {Child}(y,{Owner}(x)) \land {Bites}(x,y))$.
4. Everyone’s zip code within a state has the same first digit.
1. ${\forall\,x,s,z_1\;\;} [{State}(s) \land {LivesIn}(x,s) \land {Zip}(x)z_1] {\:\;{\Rightarrow}\:\;}{}$\ $[{\forall\,y,z_2\;\;} {LivesIn}(y,s) \land {Zip}(y)z_2 {\:\;{\Rightarrow}\:\;}{Digit}(1,z_1) {Digit}(1,z_2) ]$.
2. ${\forall\,x,s\;\;} [{State}(s) \land {LivesIn}(x,s) \land {\exists\,z_1\;\;} {Zip}(x)z_1] {\:\;{\Rightarrow}\:\;}{}$\ $[{\forall\,y,z_2\;\;} {LivesIn}(y,s) \land {Zip}(y)z_2 \land {Digit}(1,z_1) {Digit}(1,z_2) ]$.
3. ${\forall\,x,y,s\;\;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {\:\;{\Rightarrow}\:\;}{Digit}(1,{Zip}(x){Zip}(y))$.
4. ${\forall\,x,y,s\;\;} {State}(s) \land {LivesIn}(x,s) \land {LivesIn}(y,s) {\:\;{\Rightarrow}\:\;}{}$\ ${Digit}(1,{Zip}(x)) {Digit}(1,{Zip}(y))$.

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