Question: In Example, the symbol ⊕ was introduced to denote exclusive or, so p ⊕ q ≡ (p ∨ q)∧ ∼(p ∧ q). Hence the truth table for exclusive or is as follows:
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.
Example: Construct the truth table for the statement form (p ∨ q) ∧ ∼(p ∧ q). Note that when or is used in its exclusive sense, the statement "p or q" means "p or q but not both" or "p or q and not both p and q," which translates into symbols as (p ∨ q) ∧ ∼(p ∧ q). This is sometimes abbreviated p ⊕ q or p XOR q.