Challenge Level

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

$x\int^x_0 ydy < 0$ | $x> 1$ | $0< x< 1 $ | $x^2+4x+4 =0$ |

$x=0 $ | $\cos(x/2)> \sin(x/2)$ | $x> 2$ | $x=1$ |

$2\int^{x^2}_0ydy> x^2 $ | $x< 0 $ | $x^2+x-2=0$ | $x=-2 $ |

$x^3> 1$ | $|x|> 1$ | $x> 4$ | $\int^x_0 \cos y dy =0$ |

[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 $p\Rightarrow q$ and four statements $p\Leftrightarrow q$. Can you work out how to do this?

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.