How the distinction between problems (find/describe such values of x that… ) and universal truths (identities) is taught to secondary-school students and higher? Especially in English-speaking countries.
Ī mean the following thing (sorry if Ī present an obvious stuff in a confused way): if we have a relation, equality or else, and a proposition with this relation at the top and some variable(s), then we have two distinct useful cases, although notation might be the same. In the first case the proposition is not necessarily true and we have to find values of the variable where it is true. In the second case the proposition is necessarily true (law/identity/“theorem”) and variables become bound by universal quantification. Universal truths can be used in subsequent equivalent transformations of expressions, equations, inequalities, or something alike. Students must learn to distinguish these cases, haven’t they? Examples:
Binary relations: “=” (equals) “>” (greater than)
Problems: “$x^2 - x - 1 = 0$” (equation) “$x^2 - x - 1 > 0$”
Universal truths: “$(a+b)(a-b) = a^2 - b^2$” “$\exp(x) > 0$”
How may be used: “α² = δ² ⇒ α = δ ∨ α = −δ ” exp(α)⋅ℓ > 0 ⇔ ℓ > 0 (transformed equation) (transformed inequality (problem)) “(α + 1)(α − 1) = α² − 1 ” cosh α > 0 (transformed expression) (derived inequality (truth))
Why am Ī preoccupied with it? A year ago, the only person happened to cooperate with me about precise equation–identity distinction in English Wikipedia was a mathematician from France; was a coincidence he wasn’t taught in English? Today Ī argued on this topic (against two presumedly native English speakers) at a neighbouring site and become frustrated. Could an English speaker be correct in insisting that something like an+1 = a × an is an “equation” and calling the “=” symbol in it “equation sign”?