Iffy logic
Problem
This problem explores the way that mathematical statements link together logically.
However, we know that for any value of $x$ with $x> 1$ then we also have $x^2> 1$, whereas it is not always the case that if $x^2> 1$ then $x>1$ (consider $x=-2$, for example). Thus:
It is correct to write $\quad\quad x> 1 \Rightarrow x^2>1$
It is incorrect to write $\quad\quad x^2> 1 \Rightarrow x> 1$
In the interactivity below, there are sixteen statements. Assuming that $n$ and $m$ are positive integers, can you sort them into eight pairs of statements?
In four of the pairs, the implication only works in one direction, whereas in the other four pairs, each statement implies the other.
If you want to work on this away from the computer, you can print out the statements.
Some cards can be linked in more than one way, for example the following statements are both true:
- $n>0 \implies n>1$
- $n>0 \implies n>2$
However there is only one way to arrange all 16 cards so that all 8 statements are simultaneously true. Can you show that there is only one way to do this?
What happens to the statement involving $n^3>5n$ if $n$ is not necessarily positive? Are any of the other statements affected if $n$ and $m$ are not necessarily positive?
Getting Started
Some cards could be paired up with more than one possible partner.
See if you can identify cards which have an obvious partner or only match up with one other possibility (such as the two involving Rover).
There are 6 cards which involve both $n$ and $m$ - can you pair these up?
Don't forget that $n$ and $m$ are both positive integers!
Student Solutions
Edward from Aquinas College in England explained why each pair of statements fit around a $\Rightarrow$ or a $\iff$:
If Rover is a dog, he isn't a cat. If Rover's not a cat there are a lot of other things he could be.
If $2n-m<0$ then $m$ is bigger than $2n.$ It can't be the other way round: if $n\lt m,$ we don't know that $2n\lt m$
Sometimes, an example is helpful to show that something is not always true. It is not true that $n-m$ is not zero $\Rightarrow n+m$ is odd, because if $m=4$ and $n=2$ then $n-m$ is not zero, but $n+m$ is not odd.
$n$ is even, $n+1$ is odd. $even+odd=odd.$ It is reversible as $odd-odd=even$
If $n$ is bigger than $m,$ then $n-m$ will be bigger than $0$ as $n\gt m$ rearranges to $n-m\gt0$
As $n$ is a positive integer, if $n^2$ is odd then $n$ must be odd.
Similarly, if n is odd then $n^2$ is odd. As $odd\times odd=odd$
If $n\gt2$ then $n^3$ will be greater than $5n$ as $n$ has to be an integer
The nearest integer after $2$ is $3$
$3^3=27$ which is greater than $5\times3=15$
It is reversible as if $n^3\gt5n,$ $n$ will be bigger than $2$
James from KEGS in England answered the question of whether there is more than one possible solution:
To begin with, I established all possible implicative relationships using the following visual aid:
There are 4 pairs of statements which imply the other, therefore these must be the 4 pairs in the solution, indicated by 
Rover is a dog $\Rightarrow$ Rover is not a cat, there are no other connections to these two.
In the same way, $2n-m\lt0\Rightarrow n\lt m$
As with $n+m$ is odd $\Rightarrow n-m$ is not zero
This leaves the statements $n\gt1 \Rightarrow n\gt0$
Teachers' Resources
Why do this problem ?
This problem is designed to help students think clearly about logical implication. It offers an opportunity to familiarise them with the arrows $\Rightarrow$ and $\Leftrightarrow$ that they may wish to use in their own mathematical writing.
Possible approach
When introducing the problem, please emphasise to students that $n$ and $m$ are positive integers.
Students could work on laptops and tablets, or using the printable cards. The interactivity allows them to check their answers.
It is valuable for students to work on this problem in pairs or small groups, so that they can talk about their logical statements and try to explain their answers verbally. Some cards can be matched in multiple ways, so working towards a complete solution will involve swapping statements and thinking clearly about whether an implication goes one or both ways.
Key questions
Are there any statements that match with only one other statement?
Which pairs can only be linked by a $\Rightarrow$ arrow?
Which pairs can be linked by a $\Leftrightarrow$ arrow?
Possible support
Rather than attempt to match all of the cards, students could try to make a selection of true statements.
Possible extension
Invite students to think about how the constraint that $n$ and $m$ are positive integers affects their answers.
Students could then move on to IFFY Triangles and Mind Your Ps and Qs