Mind your Ps and Qs
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.
Getting Started
To do this question you need to be really sure what the two symbols $\Rightarrow$ and $\Leftrightarrow$ mean.
$p\Rightarrow q$ essentially means that IF $p$ is True THEN $q$ is true.
$p\Leftrightarrow q$ means that $p$ is true if and only if $q$ is true.
This means that $p\Rightarrow q$ and $q\Rightarrow p$.
To get started, can you arrange these eight statements into two statements of the form $p\Rightarrow q$ and two statements of the form $p\Leftrightarrow q$? Once you have done this, have a look at the remaining eight statements from the problem.
|
$x> 4$ |
$x=-2 $ | $x> 1$ | $x^2+4x+4 =0$ |
|
$x^3> 1$ |
$x^2+x-2=0$ | $x> 2$ | $x=1$ |
Student Solutions
Well done to Isaac from Glenthorne High School and Pratyush from Wilson's School in the UK, Amelie from Jersey College for Girls and Dylan who all sent in mostly correct solutions - especially to those of you who did some extra research to make sense of the $\iff$ and $\Rightarrow$ signs.
Click here to see Dylan's work, which includes Dylan's complete solution.
Isaac began:
First I labelled every statement chronologically A-O, and label the 16th statement R as P&Q are being used.
Pratyush and Amelie showed how they simplified some of the propositions, and they began matching as soon as they had simplified them. This is Patuyush's work for propositions B and K:
Note that it is not true that $x^2+x-2=0\iff x=1.$ Pratyush has already shown that $x^2+x-2=0\iff x=1\textbf{ or }x=-2.$
$x^2+x-2=0$ does not imply that $x=1,$ because $x$ could also be $-2.$ However it is true that $x=1\Rightarrow x^2+x-2=0.$
This is Amerlie's work for propositions A, F, I, M and R:
Amelie and Pratyush matched up as they went along, but Isaac used a more systematic approach (and Dylan's approach was the most systematc of all). Isaac wrote:
The only statement that was always true if A was true was J, so A$\Rightarrow$J. But if $x$ is less than $0$ then $\frac{x^3}2$ is less than $0,$ so I knew that A$\iff$J.
Repeat this process for every statement until all statements have a relationship with at least one other statement. Then I grouped together 4 $P\Rightarrow Q$ statements and 3 $P\iff Q$ statements, those being E$\Rightarrow$ R, H$\Rightarrow$K, L$\Rightarrow$F, O$\Rightarrow$G, A$\iff $J, B$\iff $M and I$\iff $N. With this we're left over with C and D, which have no shared value of $x$ thus no $P\Rightarrow Q$ or $P\iff Q$ relationship.
I focused on C, and found that the only relationship C is in with any other statement is C$\Rightarrow$F, so I replaced L$\Rightarrow$F with C$\Rightarrow$F, and compared the leftover statements L & D. L states that $x = -2,$ which is the only root to D's equation, therefore L$\iff$D, and now every statement is in a pair with 4 $P\Rightarrow Q$ relationships and 4 $P\iff Q$ relationships,
that being:
E$\Rightarrow$R, H$\Rightarrow$K, C$\Rightarrow$F, O$\Rightarrow$G,
A$\iff$J, B$\iff$M, I$\iff$N and L$\iff$D.
Teachers' Resources
These printable cards may be useful: all 16 propositions, just 8 propositions (as in the Getting Started section)
Why do this problem
This problem will help to train students in the art of careful, logical, pure thinking which will help to develop their general mathematical skill. It will require students to address issues surrounding integration, use of functions, and inequalities, without needing to go into any particular detail with calculation of integrals.
Possible approach
$$x> 4 \Rightarrow x> 3\mbox{ and } x=-2 \Leftrightarrow x^3=-8$$
are correct but
The next step is to ensure that everyone can construct their own individual examples of correct mathematical statements using propositions from the list. Once students have a couple of examples of such statements they should share them with the class and explain their reasoning. Do others agree or disagree? TALKING about such results will quickly highlight woolly or fallacious thinking and is an important part of the mathematical process.
Once the group has a feel for constructing the implications, they need to concentrate on using all of the statements to construct a complete set of 8 statements. Encourage students to consider their reasoning clearly in each case. Can the class complete the task with a clear explanation in each case?
Key questions
- What do the arrow symbols mean?
- Have you read the question carefully?
- Are there certain statements which look likely to go together in a pair?
- If an integral is positive or zero, what can we say about the area enclosed?
- What do the graphs of $\cos x$ and $\sin x$ look like?
Possible support
It is rather helpful to draw diagrams and number lines when thinking about inequalities. Shade the parts of the number line which apply to a particular inequality to help see which way round the logic flows.
If possible, start off with Iffy logic and the support materal suggested there.
The Getting Started section suggests considering half the statements (8 propositions) before looking at the remaining statements.
Possible extension
Are there multiple solutions? Can students make up a similar set of questions to give to each other to try? Can they write down really clear explanations of why, for example, $x> 4\Rightarrow x> 2$?