Logic

Single correct

• A$(\exists y)(\forall x)\,((x \ne y) \to (M(x, y) \wedge \neg M(y, x)))$
• B$(\forall y)(\exists x)\,((x \ne y) \to (M(x, y) \wedge \neg M(y, x)))$
• C$(\exists y)(\exists x)\,((x \ne y) \to (M(x, y) \wedge \neg M(y, x)))$
• D$(\forall y)(\forall x)\,((x \ne y) \to (M(x, y) \wedge \neg M(y, x)))$

One or more correct

• A$\forall x \exists y \exists z\, (mother(y, x) \wedge \neg mother(z, x))$
• B$\forall x \exists y\, [mother(y, x) \wedge \forall z\, (noteq(z, y) \to \neg mother(z, x))]$
• C$\forall x \forall y\, [mother(y, x) \to \exists z\, (mother(z, x) \wedge \neg noteq(z, y))]$
• D$\forall x \exists y\, [mother(y, x) \wedge \neg \exists z\, (noteq(z, y) \wedge mother(z, x))]$

Single correct

• A$(P(0) \wedge (\forall x\, [P(x) \Rightarrow P(x+1)])) \Rightarrow (\forall x\, P(x))$
• B$(P(0) \wedge (\forall x\, [P(x) \Rightarrow P(x-1)])) \Rightarrow (\forall x\, P(x))$
• C$(P(1000) \wedge (\forall x\, [P(x) \Rightarrow P(x-1)])) \Rightarrow (\forall x\, P(x))$
• D$(P(1000) \wedge (\forall x\, [P(x) \Rightarrow P(x+1)])) \Rightarrow (\forall x\, P(x))$

Single correct

• A$q \to \neg p$
• B$q \to p$
• C$p \to q$
• D$\neg p \to q$

One or more correct

• A$\exists x\Big(P(x) \land \forall y\,Q(x, y)\Big)$
• B$\forall x \forall y\,Q(x, y)$
• C$\exists y \forall x\Big(P(x) \implies Q(x, y)\Big)$
• D$\exists x\Big(P(x) \land \exists y\,Q(x, y)\Big)$

Answer key coming soon

Single correct

• ABoth $S_1$ and $S_2$ are tautologies.
• B$S_1$ is a tautology but $S_2$ is not a tautology.
• C$S_1$ is not a tautology but $S_2$ is a tautology.
• DNeither $S_1$ nor $S_2$ is a tautology.

Answer key coming soon

One or more correct

• A$S$ is neither a tautology nor a contradiction.
• B$S$ is a tautology.
• C$S$ is a contradiction.
• DThe antecedent of $S$ is logically equivalent to the consequent of $S$.

Answer key coming soon

Single correct

• A$\forall x(p(x) \lor W) \equiv \forall x\,p(x) \lor W$
• B$\exists x(p(x) \land W) \equiv \exists x\,p(x) \land W$
• C$\forall x(p(x) \to W) \equiv \forall x\,p(x) \to W$
• D$\exists x(p(x) \to W) \equiv \forall x\,p(x) \to W$

Answer key coming soon

Single correct

• AS1 and S2
• BS1 and S3
• CS2 and S3
• DS1, S2 and S3

Single correct

• AThere exists at least one model of $\varphi$ with universe of size less than or equal to 3.
• BThere exists no model of $\varphi$ with universe of size less than or equal to 3.
• CThere exists no model of $\varphi$ with universe of size greater than 7.
• DEvery model of $\varphi$ has a universe of size equal to 7.

Single correct

• AI only
• BI and IV only
• CII only
• DII and III only

Answer key coming soon

Single correct

• AIV only
• BI and IV only
• CII only
• DII and III only

Answer key coming soon

Single correct

• Aa tautology.
• Ba contradiction.
• Calways TRUE when $p$ is FALSE.
• Dalways TRUE when $q$ is TRUE.

Answer key coming soon

Single correct

• A$(\neg p \wedge r) \wedge (\neg r \to (p \wedge q))$
• B$(\neg p \wedge r) \wedge ((p \wedge q) \to \neg r)$
• C$(\neg p \wedge r) \vee ((p \wedge q) \to \neg r)$
• D$(\neg p \wedge r) \vee (r \to (p \wedge q))$

Answer key coming soon

Numerical answer

Answer key coming soon

Numerical answer

Answer key coming soon

Single correct

• A$(\forall x\, p(x) \Rightarrow \forall x\, q(x)) \Rightarrow (\exists x\, \neg p(x) \vee \forall x\, q(x))$
• B$(\exists x\, p(x) \vee \exists x\, q(x)) \Rightarrow \exists x\, (p(x) \vee q(x))$
• C$\exists x\, (p(x) \wedge q(x)) \Rightarrow (\exists x\, p(x) \wedge \exists x\, q(x))$
• D$\forall x\, (p(x) \vee q(x)) \Rightarrow (\forall x\, p(x) \vee \forall x\, q(x))$

Answer key coming soon

Single correct

• A$(\neg p \lor q) \land (p \lor \neg q)$
• B$(\neg p \lor q) \land (q \rightarrow p)$
• C$(\neg p \land q) \lor (p \land \neg q)$
• D$(\neg p \land \neg q) \lor (p \land q)$

Single correct

• ABoth commutative and associative
• BCommutative but not associative
• CNot commutative but associative
• DNeither commutative nor associative

Single correct

• AIf a person is known to be corrupt, he is kind
• BIf a person is not known to be corrupt, he is not kind
• CIf a person is kind, he is not known to be corrupt
• DIf a person is not kind, he is not known to be corrupt