Give a compound proposition involving propositions p, q and r that is true…
2014
Give a compound proposition involving propositions p, q and r that is true when exactly two of p, q and r are true and is false otherwise.
- A.
(p∨q∧¬r) ∧ (p∧¬q∧r) ∧ (¬p∧q∧r)
- B.
(p∧q∧¬r) ∧ (p∨q∧¬r) ∧ (¬p∧q∧r)
- C.
(p∧q∧¬r) ∨ (p∧¬q∧r) ∧ (¬p∧q∧r)
- D.
(p∧q∧¬r) ∨ (p∧¬q∧r) ∨ (¬p∧q∧r)
Attempted by 106 students.
Show answer & explanation
Correct answer: D
Correct expression: (p∧q∧¬r) ∨ (p∧¬q∧r) ∨ (¬p∧q∧r)
Why this works:
The term (p∧q∧¬r) is true exactly when p and q are true and r is false.
The term (p∧¬q∧r) is true exactly when p and r are true and q is false.
The term (¬p∧q∧r) is true exactly when q and r are true and p is false.
Because these three terms are mutually exclusive, their disjunction is true exactly when any one of the three two-true cases holds, which matches the requirement.
Alternative equivalent form: ((p∧q) ∨ (p∧r) ∨ (q∧r)) ∧ ¬(p∧q∧r)
This alternative says: at least one pair is true, but not all three, which also captures "exactly two true".