The mutual transformation of three kinds of mathematical languages

\ (\hspace{1cm}\) High School mathematics in three commonly used mathematical language: natural language, Symbolic language, graphic language, they will be in the solution of the problem is constantly transformed, if not understand their transformation, encounter the topic will be flying blind.

Example 1:

符号语言:\ (\forall x_1\in a\),\ (\exists x_2\in b\), so that the equation \ (g (x_2) =f (x_1) \) is established, first converted as follows,

符号语言:\ (\{y\mid y=f (x), x\in a\}\subseteq \{y\mid y=g (x), x\in b\}\);

自然语言: The value of the function \ (y=f (x) \) is a subset of the range of functions \ (y=g (x) \) .

Example 2:

符号语言: $ab =0\leftrightarrow $ 自然语言 :\ (a=0\) or \ (b=0\);

符号语言: $ab \neq 0\leftrightarrow $ 自然语言 :\ (a\neq 0\) and \ (b\neq0\);

符号语言: $ab \ge 0\leftrightarrow $ 自然语言 :\ (\begin{cases}a\ge 0\\b\ge0 \end{cases}\) or \ (\begin{cases}a\leq 0\\ B\leq 0 \end{cases}\);

符号语言: $ab \leq 0\leftrightarrow $ 自然语言 :\ (\begin{cases}a\ge 0\\b\leq 0 \end{cases}\) or \ (\begin{cases}a\leq 0\\b\ge 0 \end{cases}\);

符号语言: $a ^2+b^2=0\leftrightarrow $ 自然语言 :\ (a=0\) and \ (b=0\); 自然语言 :\ (A, b\) all zeros;

符号语言: $a ^2+b^2\neq 0\leftrightarrow $ 自然语言 :\ (a\neq 0\) or \ (b\neq 0\); 自然语言 :\ (A, b\) not all zeros;

Example 3:

自然语言: If the function \ (f (x) \) and function \ (g (x) \) has a symmetric point on the \ (x\) axis on the image, $\leftrightarrow $ 符号语言 : Equation \ (f ( x) =-g (x) \) has a solution;

自然语言: If the function \ (f (x) \) and function \ (g (x) \) have a symmetric point on the \ (y\) axis on the image, $\leftrightarrow $ 符号语言 : Equation \ (f (-X) =g (x) \) there is a solution;

自然语言: If the function \ (f (x) \) and function \ (g (x) \ ) has a symmetric point on the Origin \ ((0,0) \ ) on the image, $\leftrightarrow $ 符号语言 : Equation \ (f (X ) =-g (-X) \) has a solution;

Example 4:

自然语言: the solution set for \ ( x\ ) ( \cfrac{ax-5}{x-a}<0\) is \ (m\), if \ (3\in m\) and \ ( 5\notin m\), the range of realistic number \ (a\) ; $\leftrightarrow $

符号语言:\ (3\in m\) corresponds to \ (\cfrac{3a-5}{3-a}<0\),

\ (5\notin m\) corresponds to two scenarios: inequality denominator is zero \ (5-a=0\) and \ (\cfrac{5a-5}{5-a}\ge 0\),

It is necessary to solve \ (\left\{\begin{array}{l}{\cfrac{3a-5}{3-a}<0}\\{5-a=0}\end{array}\right.①\) and \ (\left\{\ Begin{array}{l}{\cfrac{3a-5}{3-a}<0}\\{\cfrac{5a-5}{5-a}\ge 0}\end{array}\right.②\)

Solution ① get \ (a=5\),

Solution ② get \ (\left\{\begin{array}{l}{\cfrac{5}{3}<a<3}\\{1\leq a<5}\end{array}\right.\), ie \ (\ \ cfrac{5}{3}<a<3\)

It is known that the range of real numbers \ (a\) is \ (\cfrac{5}{3}<a<3 or a=5\).

Example 5: When the topic appears \ (A\subseteq b\) , it often means that the collection \ (a\) has two situations:\ (a=\varnothing\) and \ (A\neq \varnothing \).

\ (A\subseteq b\) \ (\longleftrightarrow\) \ (a\cap b=a\) \ (\longleftrightarrow\) \ (A\cup b=b\) \ (\longleftrightarrow\) \ (C_ub\subseteq c_ua\)
\ (\longleftrightarrow\) \ (A\cap (c_ub) =\varnothing\)

Example 6: constant establishment and ability to set up a class proposition

① 自然语言 :\ (A\ge f (x) \) is established on the interval \ ([a,b]\) , $\leftrightarrow $ 符号语言 :\ (A\ge f (x) _{max}\) ;

自然语言:\ (A\leq f (x) \) is established on the interval \ ([a,b]\) , $\leftrightarrow $ 符号语言 :\ (A\leq f (x) _{min}\);

② 自然语言 :\ (A\ge f (x) \) can be established on the interval \ ([a,b]\) , $\leftrightarrow $ 符号语言 :\ (A\ge f (x) _{min}\);

自然语言:\ (A\leq f (x) \) can be established on the interval \ ([a,b]\) , $\leftrightarrow $ 符号语言 :\ (A\leq f (x) _{max}\);

③ 符号语言 : to \ (\forall x_1\in [2,3]\),\ (\exists x_2\in [4,5]\), satisfies \ (f (x_1) \ge g (x_2) \);

$\leftrightarrow $ 符号语言 :\ (f (x_1) _{min}\ge g (x_2) _{min}\);

④

⑤

⑥

The mutual transformation of three kinds of mathematical languages