Given this statement:
"Every positive number that is smaller than $1$ is bigger than its square"
Which of these statements are true (They may be both false/right) ?
You can write the statement as:
- $\forall x((x <1) \wedge (x >0) \wedge (x^2 <x))$
- $\forall x((x<1) \wedge (x>0)) \rightarrow \forall x(x^2 < x)$
I think that $1$ is true as it is a Tautology, but I am not sure about $2$.. ( I am not sure about $1$ as well)
I would appreciate your help! Thank you!
Neither of your statements says what is should.
I would do this: $$ \forall x\Big(\big((x>0) \wedge (x<1)\big) \rightarrow (x>x^2)\Big) $$ Or maybe $$ \forall x \Big((x>0) \rightarrow \big((x<1)\rightarrow (x>x^2)\big)\Big) $$ These are equivalent statements, and "that is" could be interpreted either way.
Note, I followed your lead and did not put in special language for "is a number" and "square".
Our target statement: "Every positive number that is smaller than $1$ is bigger than its square" is a true statement about the real numbers.
Your statement 1 $$ \forall x((x <1) \wedge (x >0) \wedge (x^2 < x)) $$ says that every number has all three properties: $x<1$ and $x>0$ and $x^2<x$. That is not what we want. This is not a true statement about the real numbers.
Your statement 2 $$ \forall x((x<1) \wedge (x>0)) \rightarrow \forall x(x^2 < x) $$ Is also not what we want. Interpreted in the real numbers, it means: if every number has both properties $x > 1$ and $x < 0$ (which is false), then every number has the property $x^2 < x$ (which is also false). This conditional statement is, in fact, true about the real numbers, but does not tell us what the target statement does.