Lets suppose we have the following:
Determine the truth value of the given statement:
If |x|=6 then x=6
Now, how does one go about determining the truth value?
It would be incorrect to use a counterexample -6 to prove this false since we are not dealing with a bi conditional 'if and only if' like this:
x>7 if and only if x>6
let x = 6.5
Then we can ge that the statement 'x>7 if and only if x>6' is false due to the left component being false and the right component being true. However in the first problem how do I determine the truth value without a given truth value of either component and with an inability to assume a truth value and determine truth/falsity via counterexample? Intuitively I know the statement is true but how do I prove this using logic?
My first attempt was to assume the first component is true and then the second component followed and was true and thus the statement was true, but this was only based upon the initial assumption that the first component was true in the first place! How do I arrive at truth value here without assumption?
Best Answer
The truth table for conditional $P \implies Q$ (you can find this online) tells us that the conditional $P \implies Q$ is false if $P$ is true and $Q$ false; otherwise it's true.
So your counterexample has to be a case where $P$ holds, but $Q$ does not.
In your example, $x=-6$ works as a counterexample. $|x|=6$ is true, but $x=6$ is false; thus $|x|=6 \implies x=6$ is false, i.e., we cannot say that whenever $|x|=6$, $x=6$.