# Logic Question

• Feb 18th 2009, 10:11 PM
RedBarchetta
Logic Question
"Another true statement about real numbers is the following: If $x^2<0$, then $x=23$."

How is this a true statement?
• Feb 18th 2009, 10:16 PM
namelessguy
Quote:

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

How is this a true statement?

I think it is true because $x^2 <0$ given $x \in R$ is false, and in mathematics if the hypothesis is false, then the statement is true no matter what the following of that hypothesis is.
• Feb 20th 2009, 07:55 PM
archidi
Quote:

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

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................................................ .........................................
• Feb 21st 2009, 01:13 PM
BoboStrategy
Re: Logic question
Quote:

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

How is this a true statement?

The statement says that for any real x such that $x^2$ is negative, x is 23. That is false if and only if there is some x so that $x^2<0$ 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