Mind Your Ps and Qs

Age 16 to 18

Challenge Level Yellow star

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.

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners

Mind Your Ps and Qs

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