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:
“The basic tool for the manipulation of reality is the manipulation of words. If you can control the meaning of words, you can control the people who must use the words.”
—Philip K. Dick (1928–1982)
“It is possible—indeed possible even according to the old conception of logic—to give in advance a description of all ‘true’ logical propositions. Hence there can never be surprises in logic.”
—Ludwig Wittgenstein (1889–1951)
“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)
“Ethical and cultural desegregation. It is a contradiction in terms to scream race pride and equality while at the same time spurning Negro teachers and self-association.”
—Zora Neale Hurston (1891–1960)