In Conjecture and Proof:

* Define something that prevents.

* Consider something smallest.

* Consider something largest.

* Find a clever coloring.

* Try using a Hamel basis.

* Try following paths.

* Count things.

In Geometric Graph Theory:

* Turn the edges into rubber bands.

* Triangulate.

* Draw circles around *everything*.

* Apply Euler’s formula.

* Pass a line across it.

* Color the edges red and blue and find the vertex where the red edges are together

* Use physics.

In Set Theory:

* Use the sandwich theorem.

* Use ridiculous seeming approximations.

* Look at the cardinality of the sets.

* Order the sets and compare ordinals.

* Consider a representative set.

* Draw the diagrams.

* Remember that everything is a set - unless it isn’t.

In Topics in Geometry:

* Try stringing definitions together.

* Try looking at equivalent definitions.

* Play with the geometry, convert to algebra, play with the algebra, convert back to geometry, repeat.

* Look for eigenvectors/values.

* Draw pictures.

* Use symmetry.

* Look at subgroups.

## 2 comments:

I'm not sure if this is what you mean by "consider a representative set", but a very important trick in set theory is to replace a set by another set with the same cardinality but different structure. For example, if you want to prove something about subsets of the natural numbers, say, it might become transparent if you consider subsets of the rational numbers instead. For a more concrete example of that, consider the statement that there exists an uncountable totally ordered (by inclusion) set of subsets of the natural numbers. This is at first rather counterintuitive, since certainly any well-ordered sequence must be countable (because at every step you have to add a new element of N). However, if you replace N by Q, it's easy: for each real r, consider A_r={q \in Q: q < r}; then {A_r} is totally ordered by inclusion and uncountable.

Other common replacements include replacing an infinite set by its square (since k^2=k for any infinite cardinal k), or an infinite set by the set of all its finite subsets.

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