Final answer:
The logical statement given is indeed a tautology because both sides of the biconditional express the same conditions: that exactly one or all three of the variables p, q, r are true, resulting in the statement being true for all possible truth assignments.
Step-by-step explanation:
The student's question involves determining whether a given logical statement is a tautology. A tautology is a statement that is true in every possible interpretation, no matter the truth values of its components. The statement in question is (p⊕q⊕r)↔((p∨q∨r)∧¬((p∧r)∨(q∧r)∨(p∧q))). To solve this, we should consider constructing a truth table for both sides of the biconditional (↔) and then checking if every possible interpretation yields the same truth value.
However, for efficiency, we'll use logical equivalences. The left side of the biconditional (p⊕q⊕r) is an exclusive or statement indicating that exactly one of the variables is true, or all three are. The right side ((p∨q∨r)∧¬((p∧r)∨(q∧r)∨(p∧q))) is a conjunction of two statements: a disjunction (p∨q∨r) indicating at least one is true, and a negated disjunction of all pairwise conjunctions which ensures no two variables are true simultaneously.
For the statement to be a tautology, it must be the case that for any truth assignment to p, q, r, the resulting truth value must always be true. We can conclude that this is the case here because both sides of the biconditional essentially encode the same condition: that there is precisely one true statement among p, q, r or that all three are true. Therefore, the statement is a tautology.