Which one of the following well formed formulae is a tautology?
(A)
∀x∃yR(x,y)<–>∃y∀xR(x,y)
(B)
(∀x(∃yR(x,y)->S(x,y)])->∀x∃yS(x,y)
(C)
[∀x∃y(P(x,y)->R(x,y)]<->[∀x∃y(~P(x,y)∪R(x,y)]
(D)
∀x∃yR(x,y)<–>∀y∀xR(x,y)
Answer: (C)
Explanation:
In logic, a tautology is a formula that is true in every possible interpretation.
All options here are based on order of application of quantifier. So, there are 2 rules:
- The positions of the same type of quantifiers can be switched
- The positions of different types of quantifiers cannot be switched.
Now, let\’s see the Choices of the question:
Option (a)
Sign <-> represents \” equivalent\”.
∀x∃y R( x, y ) is not equivalent to ∃Y ∀X R( X, Y )
Let R( X, Y ) represent X < Y for the set of numbers as the universe, for example. Then ∀X ∃Y R( X, Y ) reads \”for every number x, there is a number y that is greater than x\”, which is true, while ∃Y ∀X R( X, Y ) reads \”there is a number that is greater than every (any) number\”, which is not true. So this option is rejected.
Option (d)
Sign -> represents \”euivalent\”
∀X ∀Y R( X, Y ) is equivalent to ∀X ∀Y R( Y, X )
Let R( X, Y ) represent X < Y for the set of numbers as the universe, for example. Then ∀X ∀Y R( X, Y ) reads \”for every number X, there is every Y that is greater than x\”, while ∀X ∀Y R( Y, X ) reads \”for every number Y, there is every X that is greater than Y\”.
And both can’t be equivalent (because at one time, one will be true and other one will be false) So this option is rejected.
Option (b) is clearly rejected as two predicate can’t be equivalent to one predicate only. So Option (c) is the correct one.
Explanation for Option (c) – as position of the quantifier is not changed and since LHS P -> R = ⌐P ᴠ R which is equal to RHS. Option c is tautology and correct answer too.
Note: For solving proposition logic question, always remember not to try with rules only. Just take an example and see if options are satisfying it or not. Because for a particular example, three options will give same result and one will be different. And different one is your answer.
Quiz of this Question
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