Sufficient and Necessary Question

36) 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 answer is E yet I feel that the 'OR' needs to be an 'AND'
Why is E correct?

Hi @dontrevisefail


Wow this is hurting my head! haha

I feel that E is correct with 'or' not 'and' as in the statement is doesn't explicitly say that when bulb x is off y is also on, just that when one of them is, bulb z is on.

Initially I was drawn to A being the correct answer, but I guess that it is just the original statement in a different order, and as the new statement must 'follow from this' it would make sense that is it E as the state of bulb Z has changed from on to off.

Does this make sense? It did in my head, haha!

Matt - NTU PG TSR Rep

Thanks Matt

36) 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 answer is E yet I feel that the 'OR' needs to be an 'AND'
Why is E correct?

When you have a statement "if P then Q", then there's a logically equivalent statement called the contrapositive: "If (not Q), then (not P)".

Here, the contrapositive is "If {bulb Z is off}, then {bulb X is on and bulb Y is off}. " (which I believe is what you deduced).

So this statement does logically follow from what they give you, however it isn't on the list of options.

but: {bulb X is on and bulb Y is off} certainly does imply {bulb X is on or bulb Y is off}.

So the statement also implies E. E is a weaker condition than what could be deduced, but it still holds.