Answer to Question #227440 in Discrete Mathematics for adib

Question

Prove that the following argument is valid. Write all the necessary steps

a)Rashid didn’t perform well in the subject, but he was present in every class.

b)Every student who completed all the assignments, performed well in the subject.

c)If a student did well in the subject, then they completed all the assignments

Lead to the conclusion:

Not everyone completed all the assignments

Answer 1

$P(x): x$ perform well in the subject

$C(x): x$ was present in every class

$A(x): x$ completed all the assignment

R stands for Rashid

Premises

(1) P(R)

(2) C(R)

(3) $\forall x[A(x) \to P(x)]$

(4) $\forall x[P(x) \to A(x)]$

Conclusion

$\exist x \neg A(x)$

Steps Reason

(1) $\forall x[A(x) \to P(x)]$ Premise

(2) $\forall x[P(x) \to A(x)]$ Premise

(3) $A(R) \to P(R)$ Universal instantiation

(4) $\neg P(R)$ Premise

(5) $\neg A(R)$ Modus Tollens

(6) $\exist x \neg A(x)$ Existential generalization