edited by
13,196 views
62 votes
62 votes

Consider the following first order logic formula in which $R$ is a binary relation symbol.

$∀x∀y (R(x, y) \implies R(y, x))$

The formula is

  1. satisfiable and valid
  2. satisfiable and so is its negation
  3. unsatisfiable but its negation is valid
  4. satisfiable but its negation is unsatisfiable
edited by

7 Answers

Best answer
73 votes
73 votes

The given relation is nothing but symmetry. We have both symmetric relations possible as well as anti-symmetric but neither always holds for all sets. So they both are not valid but are satisfiable. (B) option.

edited by
24 votes
24 votes

x: boy,   y: girl
R(x,y) means x loves y.

Ok. Question says
For all boys & girls in the world, a boy loves a girl means that girl loves him too.. 
It is true sometimes too.
Hence satisfiable.
Negation of it is also satisfiable.
think logically or negate it mathematically then put this example.. In some cases these will be true.
Hence B is the answer.

edited by
2 votes
2 votes

By using Truth table and predicate formulas, this could also be another version of the answer.

1 votes
1 votes
Whenever a Predicate is satisfiable then it's negation is also satisfiable. So option B is correct.
Answer:

Related questions

42 votes
42 votes
5 answers
1
54 votes
54 votes
6 answers
2
Rucha Shelke asked Sep 18, 2014
9,119 views
Which one of the first order predicate calculus statements given below correctly expresses the following English statement? Tigers and lions attack if they are hungry or ...
3 votes
3 votes
3 answers
3
ParthPratim asked Nov 3, 2022
969 views
I have been trying to solve the question GATE CSE 2008 Question.Are the following two representations logically equivalent ?$\beta \rightarrow (\exists x, \alpha (x))$$\e...