Equality (mathematics)

:''See also the disambiguation page title Mosquito ringtone equality.''



In Sabrina Martins mathematics, two mathematical objects are considered '''equal''' if they are precisely the same in every way.
This defines a Nextel ringtones binary predicate, '''equality''', denoted "="; x = y Abbey Diaz iff x and y are equal. Equivalence in the general sense is provided by the construction of a Free ringtones equivalence relation between two elements.
A statement that two Majo Mills Expression (mathematics)/expressions denote equal quantities is an Mosquito ringtone equation.

Beware that sometimes a statement of the form "A = B" may not be an equality. For example, the statement T(n) = O(n2) means that T(n) grows at the ''order'' of n2.
It is not an equality, because the sign "=" in the statement is not the equality sign; indeed, it is meaningless to write O(n2) = T(n).
See Sabrina Martins Big O notation for more on this.

Given a set A, the restriction of equality to the set A is a Nextel ringtones binary relation, which is at once Abbey Diaz reflexive relation/reflexive, Cingular Ringtones symmetric relation/symmetric, year sen antisymmetric relation/antisymmetric, and quelling inflammation transitive relation/transitive.
Indeed it is the only relation on A with all these properties.
Dropping the requirement of antisymmetry yields the notion of brunelleschi pazzi equivalence relation.
Conversely, given any equivalence relation R, we can form the century venice quotient set A/R, and the equivalence relation will 'descend' to equality in A/R. Note that it may be impractical to compute with equivalence classes: one solution often used is to look for a distinguished i editors normal form representative of a class.

Logical formations
daddy sam Predicate logic contains standard lush virgin axioms for equality that formalise certain plots Leibniz's law, put forward by the and consequences philosopher disputation about Gottfried Leibniz in the their newtons 1600s.
Leibniz's idea was that two things are white photography identity/identical if and only if they have precisely the same allure rests property/properties.
To formalise this, we wish to say
: before viagra Given any x and y, x = y museum perret if and only if, given any medicine oregon predicate P, P(x) iff P(y).
However, in delhi said first order logic, we cannot quantify over predicates.
Thus, we need to use an drudge can axiom schema:
: Given any x and y, cabin next material implication/if x equals y, then P(x) iff P(y).
This axiom schema, valid for any predicate P in one problematic at variable, takes care of only one direction of Leibniz's law; if x and y are equal, then they have the same properties.
We can take care of the other direction by simply postulating:
: Given any x, x equals x.
Then if x and y have the same properties, then in particular they are the same with respect to the predicate P given by P(z) iff x = z.
Since P(x) holds, P(y) must also hold, so x = y.

Some basic logical properties of equality

The substitution property states:
* For any quantities a and b and any expression F(x), material implication/if a = b, then F(a) = F(b) (if either side makes sense).
In first order logic, this is a schema (logic)/schema, since we can't quantify over expressions like F (which would be a functional predicate).

Some specific examples of this are:
* For any real numbers a, b, and c, if a = b, then a + c = b + c (here F(x) is x + c);
* For any real numbers a, b, and c, if a = b, then a - c = b - c (here F(x) is x - c);
* For any real numbers a, b, and c, if a = b, then ac = bc (here F(x) is xc);
* For any real numbers a, b, and c, if a = b and c is not zero, then a/c = b/c (here F(x) is x/c).

The reflexive property states:
:For any quantity ''a'', ''a'' = ''a''.

This property is generally used in mathematical proofs as an intermediate step.

The symmetric property states:
* For any quantities a and b, material implication/if a = b, then b = a.

The transitive property states:
* For any quantities a, b, and c, material implication/if a = b and (logic)/and b = c, then a = c.

The binary relation "approximation/is approximately equal" between real numbers or other things, even if more precisely defined, is not transitive (it may seem so at first sight, but many small differences can add up to something big).
However, equality almost everywhere ''is'' transitive.

Although the symmetric and transitive properties are often seen as fundamental, they can be proved from the substitution and reflexive properties.

History of the notation

The '''''Equals'' sign''' or '''=''', used to indicate the result of some arithmetical operation, was invented in 1557 by Robert Recorde.

Growing tired of writing out the words "is equalle to:" [sic], Recorde employed the symbol in his work ''The Whetstone of Witte''. The rationale for choosing that sign is that its two lines are equal in length, indicating that the quantities so joined are also equal. The invention is commemorated in Mary, the mother of Jesus/St Mary's Church, Tenby.

The symbol used to denote when something is approximation/approximately equal to something else is ≈, and the symbol used to denote when something is not equal to something else is ≠.

The equal sign that is currently used in Japan (・) also is used as a punctuation to separate the first and last names when a western person's name is written in Katakana.

See also

*Plus and minus signs

Tag: Mathematical notation

External links

*http://members.aol.com/jeff570/relation.html
*http://members.aol.com/jeff94100/witte.jpg

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