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In a course for non-math-majors at a liberal arts college, I would like to give a few lectures and activities about groups and symmetry. I think it's straightforward to explain the group axioms and why they apply to, for instance, the set of symmetries of a regular polygon like a square. However, I would like to motivate these axioms by also presenting examples of algebraic structures that violate one, or some combination, of the standard group axioms: closure, associativity, identity, and invertibility. I also want to explain the importance of commutativity, although simple examples of noncommutative operations abound (e.g. subtraction of integers). However, the Wikipedia page for non-associativity has only technical examples that will be incomprehensible to, for instance, a History major. But that doesn't mean they can't appreciate symmetry, so ...

Q: Can you suggest particular examples of "algebraic structures", with a preference for "real life" examples, that violate some of the standard group axioms and related algebraic properties? It would be even better if you could suggest in-class activities that would make the concepts especially clear to a non-mathematically-trained student.

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    $\begingroup$ subtraction and the cross product of 3D vectors, both nonassociative. $\endgroup$ – James S. Cook Aug 14 '17 at 21:24
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    $\begingroup$ how about making friends. I can imagine a trio which exists only if the first pair makes nice before befriending the third... maybe can twist this into example for majors like history etc. $\endgroup$ – James S. Cook Aug 14 '17 at 21:39
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    $\begingroup$ See my manuscript Exotic Group Examples, which I posted at Math Forum some years ago. Although the title suggests this would not have anything immediately relevant to what you are asking for, I think you'll find quite a few examples of what you are asking for described in its annotated bibliography. $\endgroup$ – Dave L Renfro Aug 15 '17 at 14:06
  • $\begingroup$ I myself would be disinclined to give examples of failures, since my own slogan would be "everything's a group"... much like "everything's an adjoint", and so on. Not quite true, but so substantially true that it justifies (to my mind) the abstract definition. I don't want my students to worry about "checking whether something's a group", since things tend to arise naturally as visibly being groups, etc. $\endgroup$ – paul garrett Aug 15 '17 at 14:56
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    $\begingroup$ @paulgarrett Interesting point. I'm not sure that I agree in the sense that students more often than not don't really see the point of counterexamples. In Calc I they see examples which show that e.g. not all continuous functions are differentiable and not all critical points are local extrema, but they still think of differentiable and continuous as being roughly the same thing and don't bother to check if a critical number is a max or a min. They aren't in any danger of regarding "differentiable" as being misguided or fragile just because they have seen things like $|x|$ or $x^{2/3}$. $\endgroup$ – John Coleman Aug 18 '17 at 12:56
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Combining colored paint is an interesting example of a non-associative operation.

Define $Paint_1 * Paint_2$ to be the paint obtained by mixing the two paints in a $1:1$ ratio. It is easy to see that

$$(Red * Blue) * White \neq Red * (Blue * White).$$

They are different shades of purple. I haven't tried it, but it should be easy enough to make a classroom lesson out of it.

Here is a computer implementation. It is obtained by simply averaging the red, green, and blue values of the colors in 24-bit RGB color space and then using Excel VBA's RGB() function to color the interiors of various cells. I'm not sure that it completely corresponds to paint mixing, but it adequately illustrates the idea:

color mixes

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  • $\begingroup$ Excellent idea. The key is the 1:1 ratio. $\endgroup$ – Brendan W. Sullivan Aug 15 '17 at 13:26
  • $\begingroup$ Even more striking, these two colors are different from what one would get mixing Red, Blue, and White in a 1:1:1 ratio, i.e. abc with no parentheses! $\endgroup$ – Brendan W. Sullivan Aug 15 '17 at 17:33
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I really like the example of the game Rock-Paper-Scissors, thinking of it as a binary operation $\star$ on the set {Rock, Paper, Scissors}.

The rules are:

  1. Rock$\star$Paper $\mapsto$ Paper
  2. Rock$\star$Scissors $\mapsto$ Rock
  3. Paper$\star$Scissors $\mapsto$ Scissors

The operation is certainly commutative, but it is not associative. For example, $$(Rock \star Paper)\star Scissors = (Paper)\star Scissors = Scissors$$ but $$Rock \star (Paper \star Scissors) = Rock \star (Scissors) = Rock$$

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For something that's not commutative, try rotations about axes through the origin in 3D. A simple rectangular book can illustrate this very well. Try rotating the book in 90 degrees around vertical axis then 90 degrees around North-South axis, then go back to start and try in the opposite order. (Interestingly, the fact that these aren't commutative is related to the fact that multiplication in the quaternions isn't commutative.)

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I can't believe no one has mentioned (positive, say) integers under exponentiation as a non-associative operation. It's by far my favorite, though of course this example is also not commutative and has no inverses ... well, at any rate it shows why associativity isn't "obvious".

$$2^{(2^3)}\neq (2^2)^3$$

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No identity: mathematics before the discovery of zero. Addition always meant getting something new.

No inverses: burning a stick. There's no going back.

Not associative: the spread of an STD amongst three people. It certainly matters in what order you "operate".

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    $\begingroup$ hmm, sounds like a way to get to know your Title 9 officer here in the USA. That said, the students would remember this illustration. $\endgroup$ – James S. Cook Aug 15 '17 at 5:49
  • $\begingroup$ For commutativity, it can make a difference in what order you choose to marry, reproduce, and buy a house together. $\endgroup$ – Ben Crowell Aug 15 '17 at 16:55
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Some simple examples of noninvertibility:

(1) Projection. Given the object, you can determine what its shadow would look like, but the opposite is not true.

(2) Another visual example would be pixelating someone's face in an image to protect someone's identity.

(3) Dividing a natural number by 2 and throwing away the remainder.

(4) Playing musical chairs. Each round is a cyclic permutation, which is invertible, composed with eliminating one player, which is not (if we assume that the description of the state gives only a picture of the people who are still in play).

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In the game of Set, https://www.setgame.com/set, you can define a binary operation on the cards by Card1 * Card2 gives the card that completes the 'set'. (In the case that the two cards are equal, define Card1*Card1 = Card1, as this would give a 'set' according to the rules; although, in the commercially-available version of Set, there are no duplicated cards.)

This operation is commutative; is nonassociative; has no identity element, and hence inverses can't be defined. However, cancellation holds: if Card1*Card2 = Card1*Card3, then Card2 = Card3. Also if Card1*Card2 = Card3, then Card1*Card3 = Card2

Activities for this game might include practicing with the operation; convincing oneself that it gives a well-defined binary operation on the cards; and verifying (some of) the above-mentioned properties.

There is a fair amount of mathematical literature on this game. (Try googling Mathematics of the game of Set).

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Here's a "real life" example of non-commutativity: walking through a grid-like city. Starting at a point in the city, walking a block then turning left then right has you arrive at a different point from walking a block (in the same direction) then turning right then left.

For non-associativity (besides subtraction, mentioned above), division. I'm pretty sure division is useful in "real life".

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You can start with the basic operations on, say, relative integers. Multiplication is not invertible (not even when we add rationals, because of $0$), and as mentioned in a comment subtraction is not associative.

For a real-life examples (or rather non-technical close analogies, maybe) of non-commutativity, ask the students if putting on their shoes before their socks would result in a nice result (or if you are into super-heroes and not too shy, ask about putting their underwear after their pants). I am still looking for something similar about non-associativity.

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