The mathematics of equivalence reasoning

Reasoning by equivalence is is line-by-line algebraic reasoning.

Reasoning by equivalence is is an iterative formal symbolic procedure where algebraic expressions, or terms within an expression, are replaced by an equivalent until a "solved" form is reached. Reasoning by equivalence is very common in elementary mathematics. It is either the entire task (such as when solving a quadratic) or it is an important part of a bigger problem. E.g. proving the induction step is often achieved by reasoning by equivalence.

There are two modes: (i) re-writing equivalent expressions, and (ii) solving equations.

An example of working with equivalent expressions is shown below. $$\begin{array}{lll} & x^2-x-30& \\ \color{green}{\checkmark} & =x^2-2\cdot \left(\frac{1}{2}\right)\cdot x-30& \\ \color{green}{\checkmark} & =x^2-2\cdot \left(\frac{1}{2}\right)\cdot x+{\left(\frac{1}{2}\right)}^2-{\left(\frac{1}{2}\right)}^2-30 & \\ \color{green}{\checkmark} & ={\left(x-\frac{1}{2}\right)}^2-{\left(\frac{11}{2}\right)}^2& \\ \color{green}{\checkmark}&=\left(x-6\right)\cdot \left(x+5\right) \end{array}$$ It is not necessary to write the equal sign at the start of the expression in this form of reasoning.

An example of solving a quadratic equation is shown below. $$\begin{array}{cc} \ & x^2-x=30 & \\ \color{green}{\Leftrightarrow} & x^2-x-30=0 & \\ \color{green}{\Leftrightarrow} & \left(x-6\right)\cdot \left(x+5\right)=0 \\ \color{green}{\Leftrightarrow} & x-6=0\lor x+5=0 \\ \color{green}{\Leftrightarrow} & x=6\lor x=-5 \end{array}$$

STACK has predicates to determine which form of reasoning is used. The predicate is applied to the list of expressions.

1. stack_eval_arg_expression_reasoningp(ex)
2. stack_eval_arg_equation_reasoningp(ex)

Equivalent expressions

Working with equivalent expressions is relatively straightforward.

In STACK, equivalence of adjacent expressions is established with the AlgEquiv answer test.

• $\color{green}{\checkmark}$ (CHECKMARK) is use to indicate the following expression is equivalent to the previous one. E.g. $$\begin{array}{lll} & x^2-x-30& \\ \color{green}{\checkmark}&=\left(x-6\right)\cdot \left(x+5\right)\end{array}$$
• $\color{red}{?}$ (QMCHAR) is used to indicate one expression is not equivalent to the next (equation or expression).

Equivalent equations

Two equations are equivalent if they have the same solutions.

Line-by-line algebraic reasoning with equations involves different types of object:

1. Algebraic equations, which (implicitly) represent the set of solutions: e.g. $x^2+6=5x$.
2. Logical expressions giving explicit solutions: e.g. $x=2 \mbox{ or } x=3$.
3. Sets of numbers representing the solution: e.g. $\{2,3\}$, intervals such as $[0,\infty)$ (co(0,inf) in STACK).
4. Systems of inequalities, e.g. $x\geq 0$.

Reasoning by equivalence typically means working from a given equation to the explicit solutions.

In deciding whether two equations are "the same" there are a number of choices to be made.

1. Are we working over the real numbers, the complex numbers or something else?
2. What should we do about repeated solutions. E.g. are $(x-2)^3=0$ and $x=2$ equivalent equations?

There are some edge cases when reasoning with equations:

• $\mbox{false} = \{\} = \mbox{none}$ denote situations with empty solutions, such as equations such as $1=0$.
• $\mbox{true} = \mbox{all}$ denote trivial equations which are universally true, such as $x=x$. Note, the atom all is displayed as $\mathbf{R}$ rather than $\mathbf{C}$. (This can be changed with texput(all, "\\mathbb{R}");)

These edge cases differ from AlgEquiv which does not consider $\{\}$ to be $\mbox{false}$.

Note that STACK has no concept of "step size": future plans include measuring the "distance" between two expressions/equations. Teachers can then use this size to decide on whether a step is too big, or an argument needs more detail (another intermediate step).

• $\color{green}{\Leftrightarrow}$ (EQUIVCHAR) is used to indicate the following equation is equivalent to the previous one. E.g. $$\begin{array}{cc} \ & x^2-x-30=0 & \\ \color{green}{\Leftrightarrow} & \left(x-6\right)\cdot \left(x+5\right)=0\end{array}$$
• $\color{red}{?}$ (QMCHAR)is used to indicate one expression is not equivalent to the next (equation or expression).
• $\color{green}{\text{(Same roots)}}$ (SAMEROOTS) is used to indicate the same set of roots, without multiplicity.
  E.g. $${\begin{array}{lll} &x^2-6\,x=-9& \cr \color{green}{\Leftrightarrow}&{\left(x-3\right)}^2=0& \cr \color{green}{\text{(Same roots)}}&x-3=0& \cr \color{green}{\Leftrightarrow}&x=3& \cr \end{array}}$$
• $\color{red}{\Rightarrow}$ (IMPLIESCHAR) and $\color{red}{\Leftarrow}$ (IMPLIEDCHAR) are used when the solution sets of expressions are subsets. Let $P$ be the solution set of $p(x)=0$ and $Q$ be the solution set of $q(x)=0$ and $P\subset Q$ then we write $p(x) \color{red}{\Rightarrow} q(x)$. E.g. $${\begin{array}{lll} &x+1=0& \cr \color{red}{\Rightarrow}&{\left(x+1\right)}^2=0& \cr \color{red}{\Rightarrow}&{\left(x+1\right)}^3=0& \cr \color{red}{\Leftarrow}&x+1=0& \cr \end{array}}$$
• $\color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}$ is used when solving over the reals (see the option assume_real). E.g. $${\begin{array}{lll}\color{blue}{(\mathbb{R})}&x+1& \cr \color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}&{\left(x+1\right)}^2=0& \cr \color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}&{\left(x+1\right)}^3=0& \cr \color{green}{\Leftrightarrow}\, \color{blue}{(\mathbb{R})}&x+1=0& \cr \end{array}}$$
• $\color{blue}{\text{Assume +ve vars}}$ is used when solving over the positive reals (see the option assume_pos). E.g. $$\begin{array}{lll}\color{blue}{\text{Assume +ve vars}}&\left(x-7\right)\cdot \left(x+1\right)=0& \cr \color{green}{\Leftrightarrow}&x=7\,{\text{ or }}\, x=-1& \cr \color{green}{\Leftrightarrow}&x=7& \cr \end{array}$$
• $\color{green}{\log(?)}$ is used when equivalence is established using the rule $A=B \Leftrightarrow e^A=e^B$. E.g. $$\begin{array}{lll} &\log_{3}\left(\frac{x+17}{2\cdot x}\right)=2& \cr \color{green}{\log(?)}&\frac{x+17}{2\cdot x}=3^2&{\color{blue}{{x \not\in {\left \{0 \right \}}}}}\cr \color{green}{\Leftrightarrow}&x=1& \cr \end{array}$$

Other symbols are used to give feedback of various kinds.

• $\color{red}{\text{Missing assignments}}$ is used when students forget to write a variable. E.g. $${\begin{array}{lll} &x=1\,{\text{ or }}\, x=2& \cr \color{red}{\text{Missing assignments}}&x=1\,{\text{ or }}\, 2& \cr \end{array}}$$
• $\color{red}{\text{and/or confusion!}}$ is used when students use and/or incorrectly. E.g. $${\begin{array}{lll} &x=1\,{\text{ or }}\, x=2& \cr \color{red}{\text{and/or confusion!}}&\left\{\begin{array}{l}x=1\cr x=2\cr \end{array}\right.& \cr \end{array}}$$
• $\color{blue}{\int\ldots\mathrm{d}x}$ and () are used when we infer students have performed calculus operations (see the option calculus). E.g. $${\begin{array}{lll} &x^2+1& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{x^3}{3}+x& \cr \color{blue}{\frac{\mathrm{d}}{\mathrm{d}x}\ldots}&x^2+1& \cr \color{blue}{\int\ldots\mathrm{d}x}&\frac{x^3}{3}+x+c& \cr \end{array}}$$