Let # be a binary operator defined as π # π = πβ² + πβ² where π and π areβ¦
2015
Let # be a binary operator defined as
π # π = πβ² + πβ² where π and π are Boolean variables.
Consider the following two statements.
(S1) (π # π)#π = π#(π # π )
(S2) π#π = π #π
Which of the following is/are true for the Boolean variables π, π and π ?
- A.
Only S1 is true
- B.
Only S2 is true
- C.
Both S1 and S2 are true
- D.
Neither S1 nor S2 are true
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Correct answer: B
Key idea: the operator # is X # Y = X' + Y' which by De Morgan is (X AND Y)'.
Check commutativity: Q # R = (Q AND R)' = (R AND Q)' = R # Q, so the commutativity statement is true.
Check associativity (use a counterexample): let P = 1, Q = 1, R = 0. Then (P # Q) # R = ((1 AND 1)') # 0 = (1') # 0 = 0 # 0 = (0 AND 0)' = 1, while P # (Q # R) = 1 # ((1 AND 0)') = 1 # (0') = 1 # 1 = (1 AND 1)' = 0. The two results differ, so associativity fails.
Conclusion: The commutativity statement is true and the associativity statement is false, so only the commutativity statement holds.