Question

In: Computer Science

Use Boolean algebraic laws to prove the following equivalences: [ ( p → q ) ∨...

Use Boolean algebraic laws to prove the following equivalences:

  1. [ ( p → q ) ∨ ( p → r ) ] ⟷ [ p ⟶ ( q ∨ r ) ]
  2. ¬ [ ¬ ( p ∧ q ) ∧ ( p ∨ q ) ] ↔ [ ( p → q ) ∧ ( q → p ) ]

If you are able to explain some of the thought process behind the problems, that would be amazing. Thanks

Solutions

Expert Solution


Related Solutions

prove or disprove using logical equivalences (a) p ∧ (q → r) ⇐⇒ (p → q)...
prove or disprove using logical equivalences (a) p ∧ (q → r) ⇐⇒ (p → q) → r (b) x ∧ (¬y ↔ z) ⇐⇒ ((x → y) ∨ ¬z) → (x ∧ ¬(y → z)) (c) (x ∨ y ∨ ¬z) ∧ (¬x ∨ y ∨ z) ⇐⇒ ¬y → (x ↔ z)
A. Use truth tables to verify these equivalences 1. p∨p ≡ p 2. p∧p ≡ p...
A. Use truth tables to verify these equivalences 1. p∨p ≡ p 2. p∧p ≡ p 3. p∨(p∧q) ≡ p 4. p∨q ≡¬p → q 5. p∧q ≡¬(p →¬q) 6. p ↔ q ≡ (p → q)∧(q → p) B. Determine the truth value of each of these statements. (Assume the domain of variables consist of all real numbers). 1. ∃x(x2 = 2) 2. ∃x(x + 2 = x) 3. ∀x(x2 + 2 > 0) 4. ∀x(x2 = x)
Prove that (((p v ~q) ⊕ p) v ~p) ⊕ (p v ~q) ⊕ (p ⊕...
Prove that (((p v ~q) ⊕ p) v ~p) ⊕ (p v ~q) ⊕ (p ⊕ q) is equivalent to p ^ q. Please show your work and name all the logical equivalence laws for each step. ( v = or, ~ = not, ⊕ = XOR) Thank you
Prove the following set identity using logical equivalences: A ∪ (B - A) = A ∪...
Prove the following set identity using logical equivalences: A ∪ (B - A) = A ∪ B.\ (Hint: Insert a table with 2 columns and 8 rows.)
Prove the validity using laws of propositional logic and rules of inference: ∀x(P(x) → (Q(x) ∧...
Prove the validity using laws of propositional logic and rules of inference: ∀x(P(x) → (Q(x) ∧ S(x))) ∃x(P(x) ∧ R(x)) − − − − − − − − − − − − − ∴ ∃x(R(x) ∧ S(x))
Prove p → (q ∨ r), q → s, r → s ⊢ p → s
Prove p → (q ∨ r), q → s, r → s ⊢ p → s
Use the laws of propositional logic to prove that the following compound propositions are tautologies. ((?...
Use the laws of propositional logic to prove that the following compound propositions are tautologies. ((? → ?) ∧ (? → ?)) → (? → ?)
Use the laws of propositional logic to prove that the following compound propositions are tautologies. ((?...
Use the laws of propositional logic to prove that the following compound propositions are tautologies. ((? → ?) ∧ (? → ?)) → (? → ?)
Use the laws of propositional logic to prove that the following compound proposition is a tautologies...
Use the laws of propositional logic to prove that the following compound proposition is a tautologies (¬? ∧ (? ∨ ?)) → ?
Using logical equivalence laws, show that (((p v ~ q) ⊕ p) v ~p) ⊕ (p...
Using logical equivalence laws, show that (((p v ~ q) ⊕ p) v ~p) ⊕ (p v ~q) is equivalent to p v q. v = or, ~ = not, ⊕ = exclusive or (XOR). Please show the steps with the name of the law beside each step, thanks so much!
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT