Equal (math) | Wikipedia audio article

Описание: This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Equalit...)


00:00:42 1 Etymology
00:01:25 2 Basic properties
00:01:47 3 Equality as predicate
00:02:30 4 Identities
00:02:51 5 Equations
00:03:13 6 Congruences
00:03:34 7 Approximate equality
00:04:17 8 Relation with equivalence and isomorphism
00:05:00 9 Logical definitions
00:05:43 10 Equality in set theory
00:06:26 10.1 Set equality based on first-order logic with equality
00:07:09 10.2 Set equality based on first-order logic without equality
00:07:31 11 See also
00:08:14 12 Notes
00:08:57 13 References
00:09:39 14 External links
00:10:22 Relation with equivalence and isomorphism
00:11:27 Logical definitions
00:12:10 Equality in set theory
00:12:53 Set equality based on first-order logic with equality
00:13:36 Set equality based on first-order logic without equality
00:14:19 y ⇒ ∀z, (x ∈ z ⇔ y ∈ z)
00:14:40 See also



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SUMMARY
=======
In mathematics, equality is a relationship between two quantities or, more generally two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. The equality between A and B is written A = B, and pronounced A equals B. The symbol "=" is called an "equals sign". For example:




x
=
y


{\displaystyle x=y}
means that x and y denote the same object.




(
x
+
1

)

2


=

x

2


+
2
x
+
1


{\displaystyle (x+1)^{2}=x^{2}+2x+1}
means that, if x is any number, the two expressions have the same value. This may also be interpreted by saying that the two sides of the equals sign represent the same function.




{
x

|

P
(
x
)
}
=
{
x

|

Q
(
x
)
}


{\displaystyle \{x|P(x)\}=\{x|Q(x)\}}
if and only if



P
(
x
)
⇔
Q
(
x
)
.


{\displaystyle P(x)\Leftrightarrow Q(x).}
This assertion, which uses set-builder notation, means that, if the elements satisfying the property



P
(
x
)


{\displaystyle P(x)}
are the same as the elements satisfying



Q
(
x
)
,


{\displaystyle Q(x),}
then the two uses of the set-builder notation define the same set. This property often expressed as "two sets that have the same elements are equal." It is one of the usual axioms of set theory, called Axiom of extensionality.