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# Mind Your Ps and Qs

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.
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### Fixing It

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Age 16 to 18

Challenge Level

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.

A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?