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An electronic circuit contains three light bulbs, X, Y and Z, which are each either on or off at any particular time. It is known that if bulb X is off or bulb Y is on, then bulb Z is on. Which one of these statements necessarily follows from this?

A. If bulb Z is on, then bulb X is off or bulb Y is on.

B. If bulb Z is on, then bulb X is on and bulb Y is off.

C. If bulb Z is on, then bulb X is on or bulb Y is on.

D. If bulb Z is off, then bulb X is off and bulb Y is off.

E. If bulb Z is off, then bulb X is on or bulb Y is off.

F. If bulb Z is off, then bulb X is on and bulb Y is on.

The contrapositive of the given statement is If bulb Z is off, then bulb X is on and bulb Y is off. However, apparently, there is no such choice in the given choices.

QC_QAOA
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Kevin
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  • Did you try to put this into a logical expression? – Bram28 Dec 28 '18 at 14:35
  • actly the answer is e,but i dont know why – Kevin Dec 28 '18 at 14:38
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    @Kevin You correctly found the the contrapositive ....now think a little about what that says ... and you'll see how e) logically follows from that. That is, answer e) is not equivalent to the original statement, but it does logically follow from it – Bram28 Dec 28 '18 at 14:46
  • @Bram28 im sorry, could u please be more detail oriented on how does it follow from it – Kevin Dec 28 '18 at 14:46
  • @Kevin remember the logical "or" is inclusive. So, if I tell you that snow is white and grass is green, does it follow (in logic) that snow is white or grass is green? – Bram28 Dec 28 '18 at 14:46
  • oh i see,thx so much @Bram28 – Kevin Dec 28 '18 at 14:50
  • You're welcome! :) – Bram28 Dec 28 '18 at 14:54
  • I've edited your thoughts into your question (taken from the comment you posted to a wrong answer). Next time, please include such attempts in the question, so that we can better address your inquiry (and your question wouldn't have been closed). =) – user21820 Jan 01 '19 at 16:09

1 Answers1

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From the comments, I gather you correctly found the contrapositive (you should really add that to your post ... in general, you should always add your work to your post). So, that is:

If Z is off, then X is on and Y is off

Now, this is almost the same as answer e) ... except in e) you have an 'or' rather than an 'and'

But, remember that in logic the 'or' is inclusive. So, if it is true that 'P and Q', then it follows that 'P or Q'.

Likewise, if Z is off, we know X is on and Y is off. But then it is also true that X is on or Y is off. So, if Z is off, then X is on or Y is off

So, the answer is e). e) is not logically equivalent to the original statement but, as we saw, it does logically follow from it, and that is what the question asked.

Bram28
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