Propositional Logic
Q2.
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 ________.Q3.
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?Q4.
Consider the first-order logic sentence F:\forall x(\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))Q7.
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?Q8.
Which one of the following predicate formulae is NOT logically valid? Note that W is a predicate formula without any free occurrence of x.Q9.
The binary operator \neq is defined by the following truth table. Which one of the following is true about the binary operator \neq?Q10.
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))[MSQ]