Mathematical equality

In mathematics, a statement in which two expressions (same or different) denote the same mathematical object is called mathematical equality. Two mathematical objects are considered equal if the objects have the same value. For example, the phrase "the sum of two and two" and the expression "four" refer to the same mathematical object, a certain natural number. The expression "is equal to" or "is the same as" is usually represented in mathematics with the sign =. Thus, the above example is usually written as:

2+2=4{displaystyle 2+2=4,}

Source of notation

The first use of the equality sign, the equation equals modern notation 14x+15=71, taken from The Whetstone of Witte by Robert Recorde (1557).

The sign = (equals), used to indicate the result of an arithmetic operation, was devised by the mathematician Robert Recorde in 1557.

Tired of writing "is equalle to", sic, he used a pair of parallel lines, ——, in his paper Whetstone of Witte. With the publication of this book, Recorde first introduced algebra to England.`

Elementary algebra and analysis

Given three objects x, y, z, where the use of the word “object” encompasses both those present in the experience sensitive, as to the beings of reason. To indicate that two objects x and y are equal, the symbol = is used in this way:

x=and{displaystyle x=y}

This means that if two objects represented by different letters are actually the same, they are related via the equals sign.

Object Equality Axioms

Equality is defined as an equivalence relation that satisfies the following axioms:

• Reflection or identity principle: x=x,
• Symmetry: Yes x=and then. and=x,
• Transitivity: yes x=and e and=z, then x=z.
• If two symbols are equal, then one can be replaced by the other.

Properties of equality

Given a set S, endowed with the operations of addition and multiplication. If a, b, c, d are four elements in S, then for the relation of equality (=) the following properties are fulfilled:

• Yeah. a=b and c=d then.
  • a+c=b+d,
  • ac=bd:
• Cancellation of the sum: in addition to any kind of numbers, it happens that if a+c=b+c, then a=b.
• Property for cancellation of multiplication: yes ac=bc and c is not the neutral of the sum in S, then a=b.

Types

Equalities can be:

1. Conditionals or equations, in which case they are met for only some variable values, for example if 3x=6, only equality is fulfilled if x=2.
2. Identities: met for all permissible values of the variable, for example (x− − 4)2=x2− − 8x+16{displaystyle (x-4)^{2}=x^{2}-8x+16,} is an algebraic identity that is fulfilled for all values x. Another example is a function and=f(x){displaystyle y=f(x)}where the symbol x represents the independent variable, and the symbol and represents the dependent variable...

Set theory

• Two sets are equal if they have the same elements; this statement is known as axiom of extension.
• Or A = B if A is contained in B, plus B is contained in A.

An equivalence relation between the elements of a set determines on the given set a partition or a collection of equivalence classes. The set of equivalence classes is called the quotient set. We say that two elements of the original set are equivalent if they belong to the same equivalence class.

For example, the natural numbers can be divided into two classes, using the equivalence relation 'two numbers are related if they give the same remainder when divided by two'. This relationship divides numbers into two classes, even and odd. The quotient set contains two elements, which are the set of even numbers and the set of odd numbers. According to this relationship, 4 and 8 belong to the same class and are 'equivalent', but 16 and 17 belong to different classes.

Rules to fulfill a relationship ♥ ♥ {displaystyle sim ,} for equivalence:

• Reflective: x♥ ♥ x{displaystyle xsim x,}
• Simetric: Yes x♥ ♥ and{displaystyle xsim and,} then. and♥ ♥ x{displaystyle andsim x,}.
• Transitive: Yes x♥ ♥ and{displaystyle xsim and,} and♥ ♥ z{displaystyle andsim z,} then. x♥ ♥ z{displaystyle xsim z,}.

The axiom of extensionality establishes the conditions of equality between sets.

Calculation of first-order predicates with equality

Predicate logic contains the standard axioms for equality that formalize Leibniz's law, proposed by the philosopher Gottfried Leibniz in the century. XVII. Leibniz's idea was that two things are identical if and only if they have exactly the same properties. To formalize this, we must be able to say:

any x{displaystyle x,} and and{displaystyle y,}, x=and{displaystyle x=y,} Yes and only if, given any preaching P{displaystyle P,}, P(x){displaystyle P(x),} Yes and only if P(and){displaystyle P(y),}.

However, in first-order logic, we cannot quantify over predicates. Thus, we need to use an axiom scheme:

any x and andYeah. x equals and, then P(x) yes and only if P(and).

This axiom scheme, valid for any predicate P in a variable, answers only for one direction of Leibniz's law; if x and y are equal, then they have the same properties. We can guarantee the other direction simply by postulating:

given any x, x is equal to x.

Then if x and y have the same properties, then in particular they are equal with respect to the predicate P given by P (z) if and only if x = z, since P( x) holds, P(y) must also hold, so x = y depending on The variable.

The opposite relationship is a relation of difference, noted with an equal stud: I was. I was. {displaystyle neq ,}