There are 5 statements:
A. Statement B, C, or D is false.
B. If statement C is false, then statement D is true.
C. It is not true that statement E is false.
D. Statements B and C have the same truth values.
E. If statement A is true, then C is false.
There's only 1 false statement, the rest it true. Which statement is false?
The answer turned our to be A. I understand why statements C, D, and E are true, but if statement B is true then wouldn't statement D be false?
On
Statements $B$ and $E$ are implications and so whenever the if-statement is false we can't make any conclusion about the then-statement. You can think of it like this. If both $B$ and $C$ are true then the then-statement in $B$ isn't invoked.
SOLUTION: It's fairly easy to count out $B$ and $D$ as false. If one of them is false then $E$ and $A$ are true, which means $C$ is false. A contradiction. So as $D$ is true, $B$ and $C$ have the same truth value so $C$ is true too. But as all $B,C,D$ are true we have that $A$ is false.
On
No, not necessarily if you say for example:
if p is a prime greater than 3, then p is of form $6x\pm1$
This doesn't then imply, that if p is not a prime greater than 3, that it's suddenly not of form $6x\pm1$. It could be greater than three and of this form and still not be prime. 25, is of this form, and it's not prime.
On
With A false and the rest true, check the truth values of each statement.
Since A is false it means all of B,C,D are true. Consistent.
Mouse over to see the rest:
B is true: since C is true, there's no implication on truth value of D. Consistent.
C is true: says E is true. Consistent.
D is true: B and C have same truth value, namely true. Consistent.
E is true: Since A is false, there's no implication on truth value of C. Consistent
If $B = $ "If statement $C$ is false, then statement $D$ is true" happens to be true, and $C$ just happens to be true, then $C $ is not false so we can not conclude anything at all about $D$ from these to statements alone.
The only thing we can conclude is that $C$ and $D$ can not both be false. But if $C$ (or $D$) is true we can not conclude anything (from statement $B$ alone) about $D$ (or $C$)
Indeed the only way statement be can be false is if $C$ is false and $D$ is also false. Any other case then statement $B$ is true.
If $B$ were false then $C$ and $D$ are both false. If $B$ is true then $C$ and $D$ are not both false. That's all we can conclude from $B$ alone.
But $D$ says "Statements $B$ and $C$ have the same truth values". That's true if $B$ is false. But if $B$ is false then $D$ is false, so $D$ is both true and false which is impossible so $B$ must be true.
And so $C$ and $D$ can't both be false. If $C$ is false, the $B$ and $C$ are opposites and $D$ is false but $C$ and $D$ can't both be false. So $C$ is true.
So $B$ and $C$ are the same truth value so $D$ is true. So all three are true and that is no contradiction at all. As $C$ is not false, "If statement $C$ is false, then statement $D$ is true" is a true statement.
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BTW, you did not need to be told how many statements were true or false to solve this. Furthermore if either $A$ or $E$ were "Fleablood's Aunt Katherine lives in Colorado" for $A$ and "Fleablood's Aunt Katherine lives in California" for $E$. you could still solve this without knowing how many were true or false.
We know $B,C,D$ are true. $C$ says $E$ is true. So we know $B,C,D,E$ are true no matter what $E$ actually says.
What $E$ actually says is "if $A$ is true then $C$ is false". We know this is true and we know $C$ is true, so $A$ must be false know matter what $A$ says.
But if we didn't know what $E$ said, we'd have to know what $A$ said. $A$ says one of $B,C$ or $D$ is false and that isn't true so $A$ is false.
We didn't need to be told that any statements were true or false. We'd have to reach the same conclusion no matter what.