Mind Your Ps and Qs
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.
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