As I got the contrapositive of this statement, how does e) necessarily follows from this? 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.

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.