How Do Roster and Set Builder Methods Enhance Traditional Topics?

[DumpmeAdrenaline]

How do the roster method and set builder methods when combined give modern meaning to traditional topics?
For example:
Find the roots of x^2-4x+3=0
Suppose we have no knowledge of the algebraic techniques for solving this equation. Had we wished to write the solution set for this equation using the set builder method
S={x: x^2-4x+3=0}
I feel like we are being redundant. We are introducing new expressions to express the same thing. Yes the implicit form tells us what property members have in common to be part of the solution set (this is understood from the problem had we not used sets). But the task of finding the members themselves namely the implicit form of the solution set still requires us to discover techniques like factoring and the quadratic equation.

 

[Stephen Tashi]

How do the roster method and set builder methods when combined give modern meaning to traditional topics?

What do you mean by "modern meaning"? - and what "traditional topics" do have in mind? Your example show as set being defined by stating a criterion for membership and you express dissatisfaction that this criterion is not, of itself, a prescription to find all the members of the set. What do you consider "modern" or "traditional" about those facts?

 

[DumpmeAdrenaline]

Modern in the sense that the study of algebra was way before set theory. Also, that its understood without using the language of set that to find the roots of the equation if we plug an x, LHS= RHS.

 

[fresh_42]

The correct notation would be $S:=\{x\in \mathbb{R}\,|\,x^2-4x+3=0\}$. It avoids naming the actual solutions since there are polynomials that do not have explicit solutions in a closed form.

It is all about your goal.

You can write the set as $S=\{1,3\}$ or as above. Even $\mathcal{V}_\mathbb{R}(x^2-4x+3)$ will be understood.
But if the solutions were irrational numbers, would it be sufficient to list the Newton algorithm? Would $S=\{0.999 \, , \,3.001\}$ be acceptable? And if not, why? Think of solutions that cannot be expressed by formulas.

This discussion is pointless in its generality.

 

[DumpmeAdrenaline]

I am learning single variable calculus, the introduction of the course includes set theory. In the study notes its mentioned that sets may better understand the overall problem of topics in mathematics and to illustrate how the author chose an example from elementary algebra (shown above). I am not saying that what the set builder method roster methods provide are meaningless but they are expressing what's already understood that in order for x to be a root x the LHS=RHS and that we must still understand such techniques as the quadratic equation and factoring regardless of the notation employed to find the members of the set.
I think whether the solution set is acceptable or not depends on how accurate we can compute namely if we can compute up to 4 decimal digits then its not.

 

[fresh_42]

I wouldn't spend too much time on notations. It is important to understand how sets are written, how quantifiers behave, e.g. on negation, and generally how to express yourself unambiguosly.

 

[Mark44]

I wouldn't spend too much time on notations.

This...
@DumpmeAdrenaline, based on this thread and another you started, it seems to me that you are focusing more deeply on set notation than is warranted.
If the question happened to be "Find the solutions of the equation $x^2 - 4x + 3 = 0$, and you wrote $S = \{x | x^2 - 4x + 3 = 0 \}$, I doubt that many teachers would give you credit for this answer.