Propositional Logic

Let p and q be the following propositions: p: Fail grade can be given. q: Student scores more than 50% marks. Consider the statement: “Fail grade cannot be given when student score...

Let p and q be the following propositions: p : Fail grade can be given. q : Student scores more than 50% marks. Consider the statement: “Fail grade cannot be given when student sco...

Let p and q be two propositions. Consider the following two formulae in propositional logic. S 1 : (¬p ∧ (p ∨ q)) → q S 2 : q → (¬p ∧ (p ∨ q)) Which one of the following choices is...

Choose the correct choice(s) regarding the following propositional logic assertion S: S : ((P ∧ Q)→ R)→ ((P ∧ Q)→ (Q → R))

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 pleasan...

Consider the following expressions: $$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(i)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ false $$\,\,\,\,\,\,\,\,\,\,\,\,$$ $$(ii)$$ $$\,\,\,\,\,\,\,\,\,\,\,\,...

Let $$p,q,r,s$$ represent the following propositions. $$p:\,\,\,x \in \left\{ {8,9,10,11,12} \right\}$$ $$q:\,\,\,x$$ is a composite number $$r:\,\,\,x$$ is a perfect square $$s:\,...

Which one of the following Boolean expressions is NOT A tautology?

Which one of the following propositional logic formulas is TRUE when exactly two of $$p, q,$$ and $$r$$ are TRUE ?

A set of Boolean connectives is functionally complete if all Boolean function can be synthesized using those, Which of the following sets of connectives is NOT functionally complet...

$$P$$ and $$Q$$ are two propositions. Which of the following logical expressions are equivalent? $${\rm I}.$$ $${\rm P}\, \vee \sim Q$$ $${\rm I}{\rm I}.$$ $$ \sim \left( { \sim {\...

Consider the following propositional statements: $${\rm P}1:\,\,\left( {\left( {A \wedge B} \right) \to C} \right) \equiv \left( {\left( {A \to C} \right) \wedge \left( {B \to C} \...

Let $$P, Q$$ and $$R$$ be three atomic prepositional assertions. Let $$X$$ denotes $$\left( {P \vee Q} \right) \to R$$ and $$Y$$ denote $$\left( {P \to R} \right) \vee \left( {Q \t...

The following propositional statement is $$$\left( {P \to \left( {Q \vee R} \right)} \right) \to \left( {\left( {P \wedge Q} \right) \to R} \right)$$$

Let $$p, q, r$$ and $$s$$ be four primitive statements. Consider the following arguments: $$P:\left[ {\left( {\neg p \vee q} \right) \wedge \left( {r \to s} \right) \wedge \left( {...

Determine whether each of the following is a tautology, a contradiction, or neither ("$$ \vee $$" is disjunction, "$$ \wedge $$" is conjuction, "$$ \to $$" is implication, "$$\neg...

"If X then Y unless Z" is represented by which of the following formulas in propositional logic? (" $$\neg $$ " is negation, " $$ \wedge $$ " is conjunction, and " $$ \to $$ " is i...

Consider two well-formed formulas in propositional logic $$F1:P \Rightarrow \neg P$$ $$F2:\left( {P \Rightarrow \neg P} \right) \vee \left( {\neg P \Rightarrow } \right)$$ Which of...

Let $$a, b, c, d$$ be propositions. Assume that the equivalences $$a \leftrightarrow \left( {b \vee \neg b} \right)$$ and $$b \leftrightarrow c$$ hold. Then the truth value of the...

(a) Show that the formula $$\left[ {\left( { \sim p \vee Q} \right) \Rightarrow \left( {q \Rightarrow p} \right)} \right]$$ is not a tautology. (b) Let $$A$$ be a tautology and $$B...

What is the converse of the following assertion? I stay only if you go

Which one of the following is false? Read $$ \wedge $$ as AND, $$ \vee $$ as OR, $$ \sim $$ as NOT, $$ \to $$ as one way implication and $$ \leftrightarrow $$ two way implication.

Let $$p$$ and $$q$$ be propositions. Using only the truth table decide whether $$p \Leftrightarrow q$$ does not imply $$p \to \sim q$$ is true or false.

Show that proposition $$C$$ is a logical consequence of the formula $$A \wedge \left( {A \to \left( {B \vee C} \right) \wedge \left( {B \to \sim A} \right)} \right)$$ using truth t...

Which of the following is/are tautology?

Indicate which of the following well-formed formula are valid: