# Relationship of Equality

## Relationship of Equality

Relationship between two quantities that have the same value.

Relationship between two quantities that have the same value or between two representations of the same mathematical object.

### Notations

• The relationship of equality is denoted by the symbol “=”, which is read as “is equal to”.
• This symbol can only be used between numbers, numerical variables or sets.
• The relationship of inequality is denoted by the symbol “≠”, which is read as “is not equal to” or “does not equal”.
• The relationship of approximation is denoted by the symbol “≈”, which is read as “is approximately equal to”.
• For measurement conversions, the symbol “=” should be read as “is equivalent to”. For example, the relationship 1 m = 100 cm should be read as “one metre is equivalent to one hundred centimetres”.

### Properties

• The relationship of equality is reflexive, symmetric and transitive; therefore, it is a relationship of equivalence. It is also antisymmetric.
• The relationship of equality must also satisfy the following axioms:
• For all real numbers x, y and z, if x = y, then x + z = y + z (a real number may be added to each member of an equality without changing the logical value of the equality);
• For all real numbers x, y and z, if x = y, then xz = yz (a real number may be subtracted from each member of an equality without changing the logical value of the equality);
• For all real numbers x, y and z, if x = y, then xz = yz (each member of an equality can be multiplied by a real number without changing the logical value of the equality);
• For all real numbers x, y and z not equal to zero, if x = y, then x ÷ z = y ÷ z (each member of an equality may be divided by a non-zero real number without changing the logical value of the equality).

### Example

The relationship 12 + 21 = 33 is read as “twelve plus twenty-one is equal to thirty-three”.

### Educational Note

Throughout the ages and across cultures, the relationship of equality has taken on various related meanings, such as has the same value as, or has the same form and dimensions as or has the same meaning as, etc. Thus, the concepts of triangle equality, equality between equivalent relationships and equality of measures expressed in different units or systems of measurement were further developed. In short, the relationship of equality was substituted by the relationships of equivalence, congruence, similarity, etc.

In this glossary, the term equal has the meaning of identity, as in 24 = 24, or 24 = 20 + 4 (same numerical value) or a² − b² = (a + b)(ab), which expresses a numerical equality for all values of a and b. Also, two sets are equal when they contain exactly the same elements, that is, when they are duplicates of each other. Two geometric figures are equal when one fits exactly over the other (not when one is a transformation of the other, in which case they are congruent).

It is important to clearly emphasize in class the distinctions between certain relationships, such as the relationship of equality, the relationship of equivalence, the relationship of similarity, the relationship of congruence, etc., that define various properties of mathematical objects being compared.