1. Logic Question

"Another true statement about real numbers is the following: If $\displaystyle x^2<0$, then $\displaystyle x=23$."

How is this a true statement?

2. Originally Posted by RedBarchetta
"Another true statement about real numbers is the following: If $\displaystyle x^2<0$, then $\displaystyle x=23$."

How is this a true statement?
I think it is true because $\displaystyle x^2 <0$ given $\displaystyle x \in R$ is false, and in mathematics if the hypothesis is false, then the statement is true no matter what the following of that hypothesis is.

3. Originally Posted by RedBarchetta
"Another true statement about real numbers is the following: If $\displaystyle x^2<0$, then $\displaystyle x=23$."

How is this a true statement?
Let p= x^2<0 ,and q= (x=23) and by the definition of a conditional statement if a true p, implies a false q, then the statement p----->q is false. IN all the other cases p----->q is true.

Since in our case p is false irrespectively of what q is ( x could be 23 or not) p---->q is true.

The above is a semantical definition of the conditional statement:

...................p------>q................................................ .........................................

4. Re: Logic question

Originally Posted by RedBarchetta
"Another true statement about real numbers is the following: If $\displaystyle x^2<0$, then $\displaystyle x=23$."

How is this a true statement?
The statement says that for any real x such that $\displaystyle x^2$ is negative, x is 23. That is false if and only if there is some x so that $\displaystyle x^2<0$ but x is not 23. There is no such x so the statement is true (vacuously).

There is a free download of a book here, which walks through additional similar examples in detail (see the first part on logic and sets):
Bobo Strategy - Topology