Mind your Ps and Qs

Sort these mathematical propositions into a series of 8 correct statements.

Problem

Here are 16 propositions involving a real number $x$:

 

$\displaystyle x\int^x_0 y\, \mathrm{d} y < 0$

$x> 1$$0< x< 1 $$x^2+4x+4 =0$

$x=0 $

$\cos\left(\dfrac x 2\right)> \sin\left(\dfrac x 2\right)$$x> 2$$x=1$

$\displaystyle 2\int^{x^2}_0y\, \mathrm{d}y> x^2 $

$x< 0 $$x^2+x-2=0$$x=-2 $

$x^3> 1$

$|x|> 1$$x> 4$$\displaystyle \int^x_0 \cos y \, \mathrm{d}y =0$

[Note: the trig functions are measured in radians]

By choosing $p$ and $q$ from this list, how many correct mathematical statements of the form $p\Rightarrow q$ or $p\Leftrightarrow q$ can you make?

It is possible to rearrange the statements into four statements of the form $p\Rightarrow q$ and four statements of the form $p\Leftrightarrow q$. Can you work out how to do this?

 

These printable cards may be useful.

 

NOTES AND BACKGROUND

Logical thinking is at the heart of higher mathematics: In order to construct clear, correct arguments in ever more complicated situations mathematicians rely on clarity of language and logic. Logic is also at the heart of computer programming and circuitry.