Propositional Logic
Q2.
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?Q3.
Consider the following expressions: (i) false (ii) Q (iii) true (iv) P\vee Q (v) \neg Q \vee P The number of expressions given above that are logically implied by P \wedge (P \Rightarrow Q) is ________.Q4.
What is the correct translation of the following statement into mathematical logic? "Some real numbers are rational"Q5.
Let p and q be two propositions. Consider the following two formulae in propositional logic. S1: (\neg p\wedge(p\vee q))\rightarrow qS2: q\rightarrow(\neg p\wedge(p\vee q))Which one of the following choices is correct?Q6.
Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE?Q7.
Consider the following two statements. S1: If a candidate is known to be corrupt, then he will not be elected S2: If a candidate is kind, he will be elected Which one of the following statements follows from S1 and S2 as per sound inference rules of logic?Q8.
In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. You give a fair coin to a person in that room, without knowing which type he is from and tell him to toss it and hide the result from you till you ask for it. Upon asking, the person replies the following "The result of the toss is head if and only if I am telling the truth." Which of the following options is correct?Q9.
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 byQ10.
Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any free occurrence of x.