Some Basic Logical Properties of Equality
The substitution property states:
- For any quantities a and b and any expression F(x), if a = b, then F(a) = F(b) (if either side makes sense, i.e. is well-formed).
In first-order logic, this is a 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, if a = b, then b = a.
The transitive property states:
- For any quantities a, b, and c, if a = b and b = c, then a = c.
The binary relation "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, if the substitution and reflexive properties are assumed instead.
Read more about this topic: Equality (mathematics)
Famous quotes containing the words basic, logical, properties and/or equality:
“Nothing and no one can destroy the Chinese people. They are relentless survivors. They are the oldest civilized people on earth. Their civilization passes through phases but its basic characteristics remain the same. They yield, they bend to the wind, but they never break.”
—Pearl S. Buck (1892–1973)
“Nature’s law says that the strong must prevent the weak from living, but only in a newspaper article or textbook can this be packaged into a comprehensible thought. In the soup of everyday life, in the mixture of minutia from which human relations are woven, it is not a law. It is a logical incongruity when both strong and weak fall victim to their mutual relations, unconsciously subservient to some unknown guiding power that stands outside of life, irrelevant to man.”
—Anton Pavlovich Chekhov (1860–1904)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“Trying to love your children equally is a losing battle. Your children’s scorecards will never match your own. No matter how meticulously you measure and mete out your love and attention, and material gifts, it will never feel truly equal to your children. . . . Your children will need different things at different times, and true equality won’t really serve their different needs very well, anyway.”
—Marianne E. Neifert (20th century)