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I am looking for simple examples of situations where a good notation/diagram was fundamental for solving an elementary problem in mathematics (I am looking for examples accessible to a basic school student).

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    $\begingroup$ Which ages is basic school? $\endgroup$
    – Tommi
    Apr 20 at 8:43

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Two examples in probability:

  • A contingency table.
  • An events tree.
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  • $\begingroup$ And good old Venn diagrams too :-) $\endgroup$
    – fedja
    Apr 21 at 12:46
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"General interest" mathematics journals like Mathematics Magazine or The Mathematical Intelligencer sometimes have "proofs without words", and the Wikipedia article is also helpful. These are diagrams that more or less immediately prove a mathematical statement. For instance, there are a number of "rearrangement proofs" of the Pythagorean theorem. More examples are Cantone (2020) or Chakraborty (2018) or Mukherjee (2021), all of which are unfortunately paywalled. Roger Nelsen also published collections of such proofs in book form.

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Graph theory is my favorite example. Euler created it to solve the Konigsberg bridges problem. No algebra needed to play around with it.

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I think tables are a very powerful method for making sense of logic problems. Before, I knew the method, found these GRE/LSAT* problems much harder. I think the benefit is in (1) overcoming the working memory bottleneck and (2) making sure all cases are covered.

For example: https://www.youtube.com/watch?v=loyetRWK3VU

P.s. Tables are not the only form of diagram for logic problems, but are pretty common. Sometimes just linking networks are the diagrams. If you check out an LSAT prep book, they will show several types of diagrams.

*It looks like the GRE Analytical section has been changed into a writing sample. Used to consist of LSAT style questions. (Not sure if the writing sample benefits from tables/not.)

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Function Notation f(x)

Instead of:

$\forall x,y: [(x,y)\in f\implies x\in A \land y\in B]\\ \land ~ \forall x\in A: \exists y:[(x,y)\in f ~ \land ~ \forall z: [(x,z)\in f \implies z=y]]$

We can write:

$\forall x\in A: f(x)\in B$

Both fully and formally describe an arbitrary function mapping elements of a domain $A$ and to elements of a codomain $B$.

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  • $\begingroup$ Those are not notations for the same thing. The first takes pains to require $f$ to be undefined on $x$ if $x\not\in A$, and it fails to use the notation $\exists!$ which, if used, would make notating the uniqueness condition easier. Still, it could be useful when the concept of "function" is being formally defined. The second says nothing about $f(x)$ if $x\not\in A$, and it omits the uniqueness condition. $\endgroup$
    – Rosie F
    Apr 28 at 10:49
  • $\begingroup$ @RosieF (1) Neither of the above notations rules out $f$ being defined outside of $A$. They say only that $f$ is defined at least for those elements in the set $A$. (2) I think most students will find the $\exists!$ notation needlessly difficult to work with. (3) $f(x)=f(x)$ for $x\in A$. If $a, ~b\in A$ and $a=b$, then $f(a)=f(b)$, hence the uniqueness requirement is actually built into the notation. $\endgroup$
    – Dan Christensen
    Apr 29 at 4:17

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