The notation \(\exists! x P(x)\) denotes the proposition “there exists a unique \(x\) such that P(\(x\)) is true”. Give the truth values of the following statements : I. \(\exists! x P(x) \rightarrow \exists x P(x)\) II. \(\exists! x \neg P(x) \rightarrow \neg \forall x P(x)\)
Both I & II are true.
Statement I: if exactly one \(x\) makes \(P(x)\) true, then at least one such \(x\) exists, so the implication always holds.
Statement II: if exactly one \(x\) makes \(P(x)\) false, then \(P(x)\) is not true for every \(x\), so \(\neg \forall x P(x)\) holds.
Both implications are therefore valid for every domain and every predicate.
Hence, Both I & II are true.
Similar Questions
Equivalent logical expression for the Well Formed Formula (WFF),
\(\sim(\forall x)\,F[x]\)
is
Let P(m, n) be the statement “m divides n” where the Universe of discourse for both the variables is the set of positive integers. Determine the truth values of the following propositions.
(a) \(\exists m \forall n P(m, n)\) (b) \(\forall n P(1, n)\) (c) \(\forall m \forall n P(m, n)\)
Codes :
________predicate calculus allows quantified variables to refer to objects in the domain of discourse and not to predicates or functions.
The kind of symbols for basic syntactic elements of first-order logic are
A. Constant
B. Domain
C. Predicate
D. Temporal
E. Function
Choose the correct answer from the options given below:
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Let L (x, y) be the statement “x loves y” where the domain for both x and y consists of all people in the world. Use quantifiers to express “Joy is loved by everyone”.
What is the process of capturing inference process as a single inference rule?
Important Questions from First Order Logic - Teaching
Consider the statements below:
“There is a country that borders both India and Nepal.”
Which of the following represents the above sentence correctly?A knowledge base contains only \(\exists x\,AsHighAs(x,Everest)\). Apply existential instantiation by introducing a fresh witness constant. Treat Kilimanjaro solely as a new symbol with no previously assigned geographical referent.
(a) \(AsHighAs(Everest,Everest)\)
(b) \(AsHighAs(Kilimanjaro,Everest)\)
Which applications are permitted by the fresh-constant rule?
If f(x) = x is my friend, and p(x) = x is perfect. then correct logical translation of the statement “some of my friends are not perfect” is _______.
Equivalent logical expression for the Well Formed Formula (WFF),
\(\sim(\forall x)\,F[x]\)
is
Let P(m, n) be the statement “m divides n” where the Universe of discourse for both the variables is the set of positive integers. Determine the truth values of the following propositions.
(a) \(\exists m \forall n P(m, n)\) (b) \(\forall n P(1, n)\) (c) \(\forall m \forall n P(m, n)\)
Codes :
