Q3 — Engineering Mathematics

Engineering Mathematics

Q3. Marks: +2.0 UGC NET Paper 2: Computer Science 2020

Consider the following argument with premise $({\forall _x}P\left( x \right)) \vee Q\left( x \right))$ and conclusion $({\forall _x}P\left( x \right)) \wedge (\forall_xQ\left( x \right))$

(A) ∀_x (P(x) ∨ Q(x))	Premise
(B) P(c) ∨ Q(c)	Universal instantiation from (A)
(C) P(c)	Simplification from (B)
(D) ∀_x P(x)	Universal Generalization of (C)
(E) Q(c)	Simplification from (B)
(F) ∀_x Q(x)	Universal Generalization of (E)
(G) (∀_x P(x)) ∧ (∀_x Q(x))	Conjunction of (D) and (F)

1.This is a valid argument.

2.Steps (C) and (E) are not correct inferences ✓ Correct

3.Steps (D) and (F) are not correct inferences

4.Step (G) is not a correct inference

Solution

The correct answer is option2

Explanation:

For C and E to be true, the assertion should be P(c) ∧ Q(c). Hence, this is not a correct inference

Additional Information

In predicate logic, universal generalization states that if $\vdash \!P(x)$ has been derived, then $\vdash \!\forall x\,P(x)$ can be derived.

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