Question 1 |

Choose the correct choice(s) regarding the following proportional logic assertion S:

S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))

S: (( P \wedge Q) \rightarrow R) \rightarrow (( P \wedge Q) \rightarrow (Q \rightarrow R))

**[MSQ]**S is neither a tautology nor a contradiction | |

S is a tautology | |

S is a contradiction | |

The antecedent of S is logically equivalent to the consequent of S |

Question 1 Explanation:

Question 2 |

Consider the two statements.

S1: There exist random variables X and Y such that \left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2 > \textsf{Var}[X]\textsf{Var}[Y]

S2: For all random variables X and Y, \textsf{Cov}[X,Y]=\mathbb E \left[|X-\mathbb E[X]||Y-\mathbb E[Y]|\right ]

Which one of the following choices is correct?

S1: There exist random variables X and Y such that \left(\mathbb E[(X-\mathbb E(X))(Y-\mathbb E(Y))]\right)^2 > \textsf{Var}[X]\textsf{Var}[Y]

S2: For all random variables X and Y, \textsf{Cov}[X,Y]=\mathbb E \left[|X-\mathbb E[X]||Y-\mathbb E[Y]|\right ]

Which one of the following choices is correct?

Both S1 and S2 are true. | |

S1 is true, but S2 is false. | |

S1 is false, but S2 is true. | |

Both S1 and S2 are false. |

Question 2 Explanation:

Question 3 |

Let p and q be two propositions. Consider the following two formulae in propositional logic.

S1: (\neg p\wedge(p\vee q))\rightarrow q

S2: q\rightarrow(\neg p\wedge(p\vee q))

Which one of the following choices is correct?

S1: (\neg p\wedge(p\vee q))\rightarrow q

S2: q\rightarrow(\neg p\wedge(p\vee q))

Which one of the following choices is correct?

Both S1 and S2 are tautologies. | |

S1 is a tautology but S2 is not a tautology | |

S1 is not a tautology but S2 is a tautology | |

Niether S1 nor S2 is a tautology |

Question 3 Explanation:

Question 4 |

Given that

B(a) means "a is a bear"

F(a) means "a is a fish" and

E(a,b) means "a eats b"

Then what is the best meaning of

\forall x[F(x) \rightarrow \forall y(E(y, x) \rightarrow b(y))]

B(a) means "a is a bear"

F(a) means "a is a fish" and

E(a,b) means "a eats b"

Then what is the best meaning of

\forall x[F(x) \rightarrow \forall y(E(y, x) \rightarrow b(y))]

Every fish is eaten by some bear | |

Bears eat only fish | |

Every bear eats fish | |

Only bears eat fish |

Question 4 Explanation:

Question 5 |

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

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

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

\forall x(p(x)\vee W)\equiv \forall xp(x)\vee W | |

\exists x(p(x)\wedge W)\equiv \exists xp(x)\wedge W | |

\forall x(p(x)\rightarrow W)\equiv \forall xp(x)\rightarrow W | |

\exists x(p(x)\rightarrow W)\equiv \forall xp(x)\rightarrow W |

Question 5 Explanation:

Question 6 |

Consider the first-order logic sentence

\varphi \equiv \exists s\exists t\exists u\forall v\forall w\forall x\forall y\varphi (s,t,u,v,w,x,y)

where \varphi (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?

\varphi \equiv \exists s\exists t\exists u\forall v\forall w\forall x\forall y\varphi (s,t,u,v,w,x,y)

where \varphi (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. |

Question 6 Explanation:

Question 7 |

Let p, q, r denote the statements "It is raining ," It is cold", and " It is pleasant,"
respectively. Then the statement "It is not raining and it is pleasant, and it is not pleasant
only if it is raining and it is cold" is represented by

(\neg p\wedge r)\wedge (\neg r\rightarrow (p\wedge q)) | |

(\neg p\wedge r)\wedge ((p\wedge q)\rightarrow \neg r) | |

(\neg p\wedge r)\vee ((p\wedge q)\rightarrow \neg r) | |

(\neg p\wedge r)\vee (r \rightarrow (p\wedge q)) |

Question 7 Explanation:

Question 8 |

Let p, q, and r be propositions and the expression (p\rightarrowq)\rightarrowr be a contradiction. Then, the
expression (r\rightarrowp)\rightarrowq is

a tautology | |

a tautology | |

always TRUE when p is FALSE | |

always TRUE when q is TRUE |

Question 8 Explanation:

Question 9 |

Consider the first-order logic sentence F:\forall z(\exists yR(x,y)). Assuming non-empty logical
domains, which of the sentences below are implied by F?

I. \exists y(\exists xR(x,y))

II. \exists y(\forall xR(x,y))

III. \forall y(\exists xR(x,y))

IV. \neg \exists x(\forall y\neg R(x,y))

I. \exists y(\exists xR(x,y))

II. \exists y(\forall xR(x,y))

III. \forall y(\exists xR(x,y))

IV. \neg \exists x(\forall y\neg R(x,y))

IV only | |

I and IV only | |

II only | |

II and III only |

Question 9 Explanation:

Question 10 |

The statement (\neg p)\Rightarrow (\neg q) is logically equivalent to which of the statements below?

I. p\Rightarrow q

II. q \Rightarrow p

III. (\neg q)\vee p

IV. (\neg p)\vee q

I. p\Rightarrow q

II. q \Rightarrow p

III. (\neg q)\vee p

IV. (\neg p)\vee q

I only | |

I and IV only | |

II only | |

II and III only |

Question 10 Explanation:

There are 10 questions to complete.

Qno. 40. Correction in option 4

The actual option is ∀x [(tiger(x) ∨ lion(x)) → (hungry(x) ∨ threatened(x)) → attacks(x)]

At the place of “∧ , there will be ” ∨”.

Thank You Intekhab Ahmad,

We have updated the answer.

In the question 23 please update the answer. It is not 0, instead it needs to be ∀x(∃y(¬α)→∃z(¬β))

Thank You MOUNIKA DASA,

We have updated the answer.

In the question 29 please update the question. In the option iv it is not ¬∃x(¬P(x)), instead it needs to be ∃x(¬P(x))

Thank You MOUNIKA DASA,

We have updated the question.

question 18 options given has a disjunction sign.

Thank You PRAFUL Rahul,

We have updated the question.

Question no. 34 none option is correct,

Thank You dp,

We have updated the option.

Please update the b and c options of 37th question

Thank You Mounika Dasa,

We have updated the option

In question 23 please update option c .

Thank You rajeev dubey,

We have updated the answer.

in Question 9 please update

F:∀x(∃yR(x,y))