Example of properties so that "any two properties imply the third"

Question

I realise this is not a mathematics question per se, but it relates to teaching mathematics, and should have simple closed answers - therefore I have decided that MSE is an appropriate platform.

In mathematics, there are many examples of properties $\{P,Q,R\}$ so that any two imply the third. For example

• A Kähler manifold: $\{\text{Complex, Riemannian, Symplectic}\}$.
• $n$ is prime, $n$ is even, $n^n = n^2$.

My question is, are there any simple (non mathematics) examples of properties satisfying this rule? Like colours, or flavours.

One way to artificially create such a collection of three properties is to start with any two properties $\{P,Q\}$, and then define a third property $R$ to be "$P$ and $Q$". While it is certainly true that any two of $P,Q$, and $R$ will imply the third, $R$ alone will also imply the other two. The properties aren't "symmetric", and it isn't interesting. Therefore, we must add an extra condition: no property implies either of the remaining two properties.

My Question: Give three properties $\{P,Q,R\}$ (which are non-mathematical in nature) so that

1. Any two properties imply the third.
2. No single property implies any of the other two.

Thank you!

Answer 1

1. One dollar is worth $\frac67$ euro.
2. One euro is worth $\frac78$ pounds.
3. One pound is worth $\frac43$ dollars.

Answer 2

First, an example of three predicates such that any two imply the negation of the third:

In a house on the (straight) seashore, you can look out to the distance in three directions at $120°$ angles to each other. Then looking out to sea in two directions implies that you don't in the third.

Now a slightly more contrived example of three predicates such that any two imply the third:

You're trying to break out of a prison. You dug a tunnel, but apparently you miscalculated, and you've emerged in a watchtower. If it's one of the watchtowers on the corners of the prison, you could still escape by going back down and digging a bit further, but if it's the watchtower in the centre of the prison you should go back to your cell and cover your tracks or you'll be caught.

The watchtowers are round, and the outer watchtowers are centred on the corners of the rectangular prison, so that $90°$ of their circumference faces the prison and $270°$ faces the outside. You have time to make three tiny holes in the wall to get your bearings, so you decide to make them at $120°$ angles to get the best possible overview of the situation. But then after looking out onto the prison yard through two of them, you realize that any two such observations imply the third...

Answer 3

(Not nonmathematical)

In an $n$-dimensional vector space consider the statements about a subset $S$:

• $S$ is independent.
• $S$ spans.
• $S$ has cardinality $n$.