Writing symbolically, and negating

[ver_mathstats]

Homework Statement

Express each statement symbolically, including a quantification of all variables which makes the universe explicit. Negate the symbolic statement.

Every positive real number has a real square root. (Do not use the symbol √ in your solution.)

Homework Equations

The Attempt at a Solution

For this question I am unsure of how to go about it.

I assume I must use the universal quantifier because we are dealing with every positive real number.

So to write it symbolically I would do (∀x∈ℝ)(x^½>0)

Would this be correct?

Thank you.

 

[andrewkirk]

No that says that for every real number its square root is positive, which is neither correct, nor what was requested.

Try first putting the statement into words that are more similar to those used in logic.

. . . If a number is positive then it has A square root.

Then think about exactly what it means to say that a number has a square root. Hint, it will involve an existence statement about another number.

 

[fresh_42]

I think to use $\sqrt{}$ or to use the power of $\frac{1}{2}$ is meant to be the same, i.e. not allowed.
Again the trick is to use sets: Define $S_c := \{\,r\in \mathbb{R}\,:\,r^2=c\,\}$ for any $c > 0$. Then a square root of $c$ means, $S_c \neq \emptyset\,.$