GATE CSE 2020 | Mathematical Logic Question 5 | Discrete Mathematics

GATE CSE 2020

MCQ (Single Correct Answer)

-0.67

Which one of the following predicate formulae is NOT logically valid?

Note that W is a predicate formula without any free occurrence of x.

$$\forall x$$(p(x) $$ \vee $$ W) $$ \equiv $$ $$\forall x$$ p(x) $$ \vee $$ W

$$\exists x$$(p(x) $$ \wedge $$ W) $$ \equiv $$ $$\exists x$$ p(x) $$ \wedge $$ W

$$\forall x$$(p(x) $$ \to $$ W) $$ \equiv $$ $$\forall x$$ p(x) $$ \to $$ W

$$\exists x$$(p(x) $$ \to $$ W) $$ \equiv $$ $$\exists x$$ p(x) $$ \to $$ W

GATE CSE 2018

MCQ (Single Correct Answer)

-0.6

Consider the first-order logic sentence
$$\varphi \equiv \,\,\,\,\,\,\,\exists s\exists t\exists u\forall v\forall w$$ $$\forall x\forall y\psi \left( {s,t,u,v,w,x,y} \right)$$
where $$\psi $$ $$(𝑠,𝑡, 𝑢, 𝑣, 𝑤, 𝑥, 𝑦)$$ is a quantifier-free first-order logic formula using only predicate symbols, and possibly equality, but no function symbols. Suppose $$\varphi $$ has a model with a universe containing $$7$$ elements.

Which one of the following statements is necessarily true?

There exists at least one model of $$\varphi $$ with universe of size less than or equal to $$3.$$

There exists no model of $$\varphi $$ with universe of size less than or equal to $$3.$$

There exists no model of $$\varphi $$ with universe of size greater than $$7.$$

Every model of $$\varphi $$ has a universe of size equal to $$7.$$

GATE CSE 2016 Set 2

MCQ (Single Correct Answer)

-0.6

Which one of the following well-formed formulae in predicate calculus is NOT valid?

$$\left( {\forall xp\left( x \right) \vee \forall xq\left( x \right)} \right) \Rightarrow \left( {\exists x\neg p\left( x \right) \vee \forall xq\left( x \right)} \right)$$

$$\left( {\exists xp\left( x \right) \vee \exists xq\left( x \right)} \right) \Rightarrow \exists x\left( {p\left( x \right) \vee q\left( x \right)} \right)$$

$$\exists x\left( {p\left( x \right) \wedge q\left( x \right)} \right) \Rightarrow \left( {\exists xp\left( x \right) \wedge \exists xq\left( x \right)} \right)$$

$$\forall x\left( {p\left( x \right) \vee q\left( x \right)} \right) \Rightarrow \left( {\forall xp\left( x \right) \vee \forall xq\left( x \right)} \right)$$

GATE CSE 2015 Set 2

MCQ (Single Correct Answer)

-0.6

Which one of the following well formed formulae is a tautology?

$$\forall x\,\exists y\,R\left( {x,y} \right) \leftrightarrow \exists y\forall x\,R\left( {x,y} \right)$$

$$\left( {\forall x\left[ {\exists y\,R\left( {x,y} \right) \to S\left( {x,y} \right)} \right]} \right) \to \forall x\exists y\,S\left( {x,y} \right)$$

$$\left[ {\forall x\,\exists y\,\left( {P\left( {x,y} \right)} \right. \to R\left( {x,y} \right)} \right] \leftrightarrow \left[ {\forall x\,\exists y\,\left( {\neg P\left( {x,y} \right)V\,R\left( {x,y} \right)} \right.} \right]$$

$$\forall x\,\forall y\,P\left( {x,y} \right) \to \forall x\forall y\,P\left( {y,x} \right)$$