Prove or disprove: There does not exist a real number x such that for all real numbers y, xy=1.
Contradiction: Suppose yes. That is, suppose that there is a real number x such that for all real numbers y, x*y=1.
Let x be the real number 0 and let y=z for any real number z.
x*y= 0*z .........by substitution
= 0 ......... by algebra
There is a contradiction, because when x is the real number 0, x multiplied by y equals 0. Therefore, the original statement is in fact true. There does not exist a real number x such that for all real numbers y, xy=1. **end of proof**
The only concern w/ my proof: Isn't the statement a universal statement? I thought that you are not allowed to use numbers to PROVE a universal statement using actual numbers, but this is the only proof I was able to come up with. Also, isn't this prove more of a counterexample as opposed to a contradiction?