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Suppose $A, B,$ and $C$ are statements such that $C$ is true if exactly one of $ A$ and $B$ is true. If $C$ is false which of the following statements must be true?

A) if $A$ is true, then $B$ is false.

B) if $A$ is false, then $B$ is false.

C) if $A$ is false, then $B$ is true.

D) both $A$ and $B$ are true.

E) both $A$ and $B$ are false.

I have two questions. First question:

$C$ is true if EXACTLY one of $A,B$ then the last two choices are right?

second question what is the difference between C) and D) ?

IrbidMath
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2 Answers2

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If C is false then either both A and B are true or false. You don't know which it is for certain, so options D and E are out.

However, if A is true then B must be true as well, otherwise C would be true. If A were false then B must be false for the same reason. You can also exchange A and B and the same argument holds.

Joel
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Answer is E. Because true of C create three options:

  • C true if A is true but B false

  • C true if A is false but B true

  • C true if both A and B true.

So the answer will be E if both A and B are false then C will be false.

sayed.allamah.iqbal
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  • The third is not correct. If both of A and B are true then C is FALSE since to get C true EXACTLY one of them should be true – IrbidMath Feb 19 '17 at 17:08