ee20-hw2-s08-sol - EECS 20N Structure and Interpretation of...

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EECS20N:StructureandInterpretationofSignalsandSystems Department of Electrical Engineering and Computer Sciences U NIVERSITY OF C ALIFORNIA B ERKELEY Problem Set 2 SOLUTIONS HW2.1 (a) DeMorgan’sLaws P Q ¬ ( P Q ) ¬ P ∨ ¬ Q T T F F T F T T F T T T F F T T P Q ¬ ( P Q ) ¬ P ∧ ¬ Q T T F F T F F F F T F F F F T T (b) DistributiveLaws P Q R Q R P ( Q R ) T T T T T T T F T T T F T T T T F F F F F T T T F F T F T F F F T T F F F F F F P Q R P Q P R ( P Q ) ( P R ) T T T T T T T T F T F T T F T F T T T F F F F F F T T F F F F T F F F F F F T F F F F F F F F F 1
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P Q R Q R P ( Q R ) T T T T T T T F F T T F T F T T F F F T F T T T T F T F F F F F T F F F F F F F P Q R P Q P R ( P Q ) ( P R ) T T T T T T T T F T T T T F T T T T T F F T T T F T T T T T F T F T F F F F T F T F F F F F F F (c) From lecture, ( P unless Q ) ( ¬ Q P ) . P Q ¬ Q P P or Q T T T T T F T T F T T T F F F F (d) From lecture, ( P unless Q ) ( ¬ Q P ) . P Q ¬ Q P P or Q T T T T T F T T F T T T F F F F HW2.2 (a) P must be false. Consider the truth table: 2
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P Q P Q P ⇒ ¬ Q T T T F T F F T F T T T F F T T Consequently, for the two implications to be true at the same time, P must be false. (b) We cannot conclude that S is linear. The phrase “System S is not linear, unless it is ZIZO“ is codified by the implication: ¬ S ZIZO ⇒ ¬ S L which does not say anything about what happens when S ZIZO is true.
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