Examples of solving for unknowns using equivalence relations that are not equality, inequality, or boolean truth?

In a book i'm writing, i want to introduce students to equations in slightly more general terms. Solving an equation with some unknown is just one example of finding an object by fact that it is a part of a certain equivalence class - specifically a class in the "equals"-relation. Same goes for inequalities too. In both scenarios we use various operations on both sides, make sure that none of these operations change the equivilance class of any member, and then isolate some desired expression. This is really intuitive and natural to work with for equality, but i think it could be a great exercise for the mind and maturity to try something very different. Does there exist any other nice similair problems and techniques for different relations? Maybe something using congruence or similarity of figures? We can obviously work with set equalities too, though that may be a little very abstract.

For information, my readers has not yet at this point in the book learned calculus. It's at just the point before. The target audience is math-interested high-schoolers.

• An equation is a statement of equality. Using it to mean something else is only going to lead to confusion. – Peter Taylor Jul 25 '19 at 7:25
• An equation is a statement of equality in terms of an equivalence relation. So, a general question could be: given two objects, do they belong to the same equivalence class? – SCS Jul 25 '19 at 9:08
• Even after your edit, I still don't understand exactly what you're talking about. You write that solving inequalities involves "mak[ing] sure that none of these operations change the equivalance class of any member," but what is the equivalence relation you're talking about when you say this? The phrase "equivalence class" is completely meaningless outside of the context of an equivalence relation. (The title doesn't make sense either, since it says "equivalence relations that are not equality, inequality or boolean truth," but inequality and boolean truth are not equivalence relations.) – Tanner Swett Jul 25 '19 at 17:29
• In general terms, what you are asking is about different types of relations between entities, of which equations are a subgroup. Inequalities are not equations, and most of the suggestions below are not equations. Please correct your post accordingly? – amWhy Jul 25 '19 at 17:42
• Also, I can't think of how solving an equation can be construed as "finding an object by fact that it is a part of a certain equivalence class". If I solve the equation $x^2 = 4$, I'm finding some objects by the fact that they're elements of a certain set, but that set is not an equivalence class. – Tanner Swett Jul 25 '19 at 17:43

Maybe proportions. Granted there's an equality in there but the emphasis is on proportions. You can even generalize the idea to SAT analogies.

Perhaps conversions or dimensional analysis would fit well in the book. Another idea is the piano tuner business case, estimating methods ala Fermi. Note these are not strictly relations. But might fit well into what you are trying to do and would work with the audience.

Maybe also some simple probability stuff with application to gambling and cards and dice and such. Things like 4:1 odds means 20% win chance.

A little bit of finance math nice also. Compound interest. Time value of money to convert between future and present values.

• I am confused. In a gambling game, if I am give 4 to 1 odds, doesn't that mean my chances of winning are 25%? (assume zero to the house) – JTP - Apologise to Monica Jul 25 '19 at 11:44
• 4:1 odds mean your probability is $\frac{1}{4+1}$, not $\frac14$. For example even odds are 1:1, which corresponds to a probability of $\frac{1}{1+1}$. Oh -- I see. The answer originally said 80%. I'll edit to 20%. @JoeTaxpayer – Chris Cunningham Jul 25 '19 at 12:27
• Ha. Ok, thanks. Now it makes sense. A $4 return on a$1 bet is not “4:1” odds. I think I understood the math but not the way of articulating it correctly. Thanks again. – JTP - Apologise to Monica Jul 25 '19 at 12:34

Consider the equivalence class of knot diagrams depicting the unknot. Given a knot diagram, Reidemeister moves do not change the knot type. So one can apply these moves to a knot diagram because "none of these operations change the equivalence class." If you reach the unknot, then you know your original diagram was just a different drawing of the unknot.

Fig: Dominic Goulding, "Knot Theory: The Yang-Baxter Equation, Quantum Groups and Computation of the Homfly Polynomial," 2010.

Also, let's not forget isomorphisms in group theory,

nor congruence relations in geometry, and, e.g., similarity in geometry. Further, in geometry, relations between lines might include $$\overline{AB} \parallel\, \overline{CD}$$, or $$\overline{EF} \perp \overline{GH}$$ (two lines being parallel, or two lines being perpendicular, respectively

There are also also congruence equations,$$\mod n: \;\; 3 \equiv 10 \pmod 7$$.

• Isomorphisms are probably going to be too advanced for the intended audience. – Jessica B Jul 25 '19 at 20:59