Equation

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Equation(Toukiki) is to express the relationship between two objects - equality.

table of contents

Introduction

The equation is to combine the two objects _ a _, _ b _ by the symbol "=", called equal sign

a = b {\ displaystyle a = b}! [a = b] (https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4)

It is written as follows. At this time, _a _ and _ b _ are (equal to each other)equal, (phase)equal,equal It is said. Also, the target corresponding to _a _ is the left sideof the equation, the object corresponding to _ b _ is called theright sideof the equation and the left side and the right side are taken as a whole Both sides, each calledeach side. Also, this denial

a ≠ b {\ displaystyle a \ neq b}! [a \ neq b] (https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf17ce351d08f2b7a3e54ba3aa2132f260c84f6)

And _a _ and _ b _ are not equal, ordifferent. Symbol "≠" isEqual sign negation It is called. The equality relationship indicated by the equation must satisfy all the following conditions as a binary relation:

: Whatever the target _a _ is, _ a _ = _ a _ always holds. : When a _ = _ b _ is established for target _ a , _ b , _ b _ = _ a _ also holds at the same time. : When a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . Substitution Principle: When target _a , _ b _ is _ a _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ As long as it has meaning) always holds.

It can be said that this is equivalent to saying that "equality is satisfying the substitution principle", and that equations give the most basic equivalence relation in mathematics.

Here, it is sometimes apparent that the things that are different are equal, because something apparently looking the same from the viewpoint of notation sometimes points to another object, so for something to be equal to each side It is necessary to be aware that you must clarify what you are targeting or who you are targeting. In some cases, even if they are not equal with each other, they are called ",,", etc., and they may use unique symbols instead of equal signs.

The substitution principle is a little more general, target a _ i , _ b _ j _

\ a_ {1} = b_ {1}, \ a_ {2} = b_ {2}, \ \ ldots, \ a_ {l}, b = = b_ {l}}! a_ {1} = b_ {1}, \ a_ {2} = b_ {2}, \ \ \ ldots, \ a_ {l} = b_ {l}] (https : //wikimedia.org/api/rest_v1/media/math/render/svg/ba80c2cad541f7b3876dc5c5c7ecf2f833c9a40a)

, Any P (_ x _ 1, _ x _ 2, …

x _ l _)

{\ Displaystyle P (a_ {1}, a_ {2}, \ ldots, a_ {l}) = P (a 1, a 2, P_ (a_ {1}, a_ {2}, \ \ ldots, a_ {l} ) = P \ (b_ {1}, b_ { {1}, b_ {2}, \ ldots, b_ {l} )] (https://wikimedia.org/api/rest_v1/media/math/render/svg/83048cae481033ab784a9dee439309beacc682d0)

There are cases to mention it in the form that it holds. This appears in the proposition P (a ) when the free variable x _ appears multiple times in _P _ ( x ) It implies that part of _a _ may be replaced by that equivalent. Because, for all ___ _ _ _ _ _ _ _ , some For j _ _ _ _ _ _ _ _ and j _ for other _b _ _ _ _ _ _ _ and put it.

arithmetic

For a , _ b , and _ c _ as arbitrary constants, when _ a _ _ _ _

• _ a _ \ + _ c _ = _ b _ \ + _ c _,
• _ a _ - _ c _ = _ b _ - _ c _,
• _ ac _ = _ bc _,
• _ a _ / _ c _ = _ b _ / _ c _

As long as both sides can be defined. This is derived from the substitution principle by the expression x _ ± c , _ x c , _ x _ / _ c . In particular,

a _ _ _ _ c _

It will be in the same order

a _ - (± _ c ) = _ b _

It is equivalent to being equal to and follows. This seems to appear as an operation of shifting some of the terms on one side to the other side while changing the sign, so this equivalent 2 Replacing one of the expressions with the other is called atranspose(transpose).

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External link

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Acquired from ""

Post Date : 2018-03-01 19:00

This article is taken from the Japanese Wikipedia Equation

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