Question

Express each of these statements using logical operators, predicates, and quantifiers.

a) Some propositions are tautologies.

b) The negation of a contradiction is a tautology.

c) The disjunction of two contingencies can be a tautology.

d) The conjunction of two tautologies is a tautology.

Short answer

a) "Some propositions are tautologies”is a statements using logical operators, predicates, and quantifiers is express in the way of $\exists \mathbf{x}\mathbf{\left(}\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right)}$

b) " The negation of a contradiction is a tautology”is a statements using logical operators, predicates, and quantifiers is express in the way of $\mathbf{\exists }\mathbf{x}\left(P\left(x\right)\right)$

c)"The disjunction of two contingencies can be a tautology”is a statement using logical operators, predicates, and quantifiers is express in the way of$\exists \mathbf{x}\mathbf{\exists }\mathbf{y}\mathbf{\left(}\mathbf{\right(}\mathbf{R}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{R}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{)}\mathbf{}\mathbf{\to }\mathbf{(}\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\vee }\mathbf{y}\mathbf{)}\mathbf{\left)}\mathbf{\right)}$

d)"The conjunction of two tautologies is a tautology"is a statement using logical operators, predicates, and quantifiers is express in the way of $\mathbf{\forall }\mathbf{x}\mathbf{\forall }\mathbf{y}\mathbf{\left(}\mathbf{\right(}\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{P}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{(}\mathbf{P}\mathbf{(}\mathbf{x}\mathbf{\wedge }\mathbf{y}\mathbf{)}\mathbf{\left)}\mathbf{\right)}$

Step-by-step solution

∃x(P(x))

Let the domain be the collection of all propositions.

Let $\mathbf{P}\left(x\right)$mean “x is a tautology".

∃x(P(x))

Let the domain be the collection of all propositions.

LetP(x)mean “xis a tautology" and letQ(x) mean “xis a contradiction".

$\mathbf{\neg }\mathbf{x}$is the negation of the proposition x

Let the domain be the collection of all propositions.

LetP(x)mean “xis a tautology", $\mathbf{R}\left(x\right)$mean “is a contingency"

∀x∀y((P(x)∧P(y))→(P(x∧y)))

Let the domain be the collection of all propositions.

LetP(x)mean “xis a tautology".