Thread: Determine whether these are valid arguments

1. Determine whether these are valid arguments

1) If $x$ is a positive real number, then $x^2$ is a positive real number. Therefore, if $a^2$ is positive, where $a$ is a real number, then $a$ is a positive real number.
2) If $x^2 /= 0$, where $x$ is a real number, then $x /= 0$. Let a be areal number with $a^2 /= 0$; then $a /= 0$

2. Originally Posted by ilovemymath
1) If $x$ is a positive real number, then $x^2$ is a positive real number. Therefore, if $a^2$ is positive, where $a$ is a real number, then $a$ is a positive real number.
This is not valid. $\displaystyle a^2 \implies a = \pm\sqrt{a^2}$

3. Originally Posted by ilovemymath
2) If $x^2 /= 0$, where $x$ is a real number, then $x /= 0$. Let a be areal number with $a^2 /= 0$; then $a /= 0$
The words "if", "then", "where", etc. don't cease to have their usual meanings because they are used in a mathematical context. Consider this argument: "If x has gills, where x is a vertebrate, then x lives in water. Let Blinky be a vertebrate with gills; then Blinky lives in water."

4. This is not valid since a^2 is positive whether a is positive or negative...