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Discrete Mathematics Proof: ∀x(P(x)∨Q(x)) → ∀x(P(x) → R(x) ∨ Q(x)), Exercises of Mathematics

2nd Military Medical UniversityMathematics

A proof using the rules of propositional logic to establish the conditional statement: for all x, (p(x) ∨ q(x)) → (p(x) → r(x) ∨ q(x)). The proof is based on the given premises: ∀x(p(x)∨q(x)) and ∀x((¬p(x) ∧ q(x))→r(x)). The proof proceeds through a series of logical steps, including universal instantiation, logical equivalence, de morgan's law, double negation law, commutative law, associative law, resolution, and idempotent law.

Typology: Exercises

2019/2020

Uploaded on 05/20/2020

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Discrete Mathematics
1.6.28
Step/Reason
1. ∀x(P(x)∨Q(x))/Premise
2. ∀x((¬P(x) ∧ Q(x))→R(x))/Premise
3. P(c)∨Q(c)/Universal instantiation from (1)
4. (¬P(c) ∧ Q(c))→R(c) /Universal instantiation from (2)
5. ¬(¬P(c) ∧ Q(c))∨R(c)/Logical equivalence from (4)
6. (¬¬P(c) ∨¬Q(c) )∨R(c)/De Morgan’s law from (5)
7. (P(c) ∨¬Q(c)) ∨R(c)/Double negation law from (6)
8. (¬Q(c) ∨P(c)) ∨R(c)/Commutative law from (7)
9. ¬Q(c) ∨(P(c)∨R(c))/Associative law from (8)
10. P(c) ∨(P(c)∨R(c))/Resolution from (3) and (9)
11. (P(c) ∨P(c)) ∨R(c)/ Associative law from (10)
12.P(c) ∨R(c)/Idempotent law from (11)
13.R(c) ∨P(c)/Commutative law from (12)
14. ¬R(c) →P(c)/ Logical equivalence from (13)
15. ∀x(¬R(x)→P(x))/Universal generalization from (14)

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Discrete Mathematics 1.6. Step/Reason

  1. ∀x(P(x)∨Q(x))/Premise
  2. ∀x((¬P(x) ∧ Q(x))→R(x))/Premise
  3. P(c)∨Q(c)/Universal instantiation from (1)
  4. (¬P(c) ∧ Q(c))→R(c) /Universal instantiation from (2)
  5. ¬(¬P(c) ∧ Q(c))∨R(c)/Logical equivalence from (4)
  6. (¬¬P(c) ∨¬Q(c) )∨R(c)/De Morgan’s law from (5)
  7. (P(c) ∨¬Q(c)) ∨R(c)/Double negation law from (6)
  8. (¬Q(c) ∨P(c)) ∨R(c)/Commutative law from (7)
  9. ¬Q(c) ∨(P(c)∨R(c))/Associative law from (8)
  10. P(c) ∨(P(c)∨R(c))/Resolution from (3) and (9)
  11. (P(c) ∨P(c)) ∨R(c)/ Associative law from (10) 12.P(c) ∨R(c)/Idempotent law from (11) 13.R(c) ∨P(c)/Commutative law from (12)
  12. ¬R(c) →P(c)/ Logical equivalence from (13)
  13. ∀x(¬R(x)→P(x))/Universal generalization from (14)