Which one of the following predicate formulae is NOT logically valid?

Note that W is a predicate formula without any free occurrence of x.

A.

∀x(p(x) ∨ W) ≡ ∀xp(x) ∨ W

B.

∃x(p(x) ∧ W) ≡ ∃xp(x) ∧ W

C.

∀x(p(x) → W) ≡ ∀xp(x) → W

D.

∃x(p(x) → W) ≡ ∀xp(x) → W

Solution:

'W' is a formula, without any free occurrence of 'x' means there is no effect of quantifiers on 'W'

Option (A): ∀x(p(x) ∨ W) ≡ ∀xp(x) ∨ W

→ Valid [From the above statement]

Option (B): ∃x(p(x) ∧ W) ≡ ∃xp(x) ∧ W

→ Valid [From the above statement]

Option (C)

Let W is “False”

P(x): ‘x’ has passed the physics exam

Let ‘x’ be the domain of students 

RHS

\forall \mathrm{xp}(\mathrm{x}) is false

False → False is True

LHS

LHS ≠ RHS.  

LHS =FALSE RHS=TRUE

RHS → LHS is False. ( "A" is equivalent to "B" where A and B are formulas iff A<=>B holds true )

Hence not valid.