Answer to Question #171462 in Discrete Mathematics for Zeshan

Question

a) Find simpler statement forms that are logically equivalent to 𝑝 ⊕ 𝑝 and (𝑝 ⊕ 𝑝) ⊕ 𝑝.

b) Is (𝑝 ⊕ 𝑞) ⊕ 𝑟 ≡ 𝑝 ⊕ (𝑞 ⊕ 𝑟)? Justify your answer.

c) Is (𝑝 ⊕ 𝑞) ∧ 𝑟 ≡ (𝑝 ∧ 𝑟) ⊕ (𝑞 ∧ 𝑟)? Justify your answer. 

Answer 1

a) Find simpler statement forms that are logically equivalent to 𝑝 ⊕ 𝑝 and (𝑝 ⊕ 𝑝) ⊕ 𝑝.

𝑝 ⊕ 𝑝=0

(𝑝 ⊕ 𝑝)⊕ 𝑝 = 0⊕ 𝑝 = 𝑝

b) Is (𝑝 ⊕ 𝑞) ⊕ 𝑟 ≡ 𝑝 ⊕ (𝑞 ⊕ 𝑟)? Justify your answer.

(𝑝 ⊕ 𝑞) ⊕ 𝑟 ≡ 𝑝 ⊕ (𝑞 ⊕ 𝑟) because of the folowing truth table

c) Is (𝑝 ⊕ 𝑞) ∧ 𝑟 ≡ (𝑝 ∧ 𝑟) ⊕ (𝑞 ∧ 𝑟)? Justify your answer.

The statement is not correct according to the folowing truth table