Download What is a rational number? We`d like to define a rational number as

What is a rational number?
We’d like to define a rational number as a number that can
be written in the form a/b, for integers a and b. But there’s a
problem with this definition. It’s circular, because it assumes the
number we are trying to define already exists.
The next try is to define a rational number to be simply the
pair of integers (a, b). But there is a problem with this too,
because the rational numbers 1/2 and 2/4 are equal, but the
pairs (1, 2) and (2, 4) are different. This leads us to the idea
of an equivalence relation. For rational numbers, we’d like to
consider two pairs (a, b) and (c, d) as being equivalent if ad = bc
(because that’s the rule you get for equivalent fractions by cross
multiplying).
First we need to say what a relation is.
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Relations
A relation between two sets A and B is a subset R ⊂ A × B. This
is like the definition of a function: if the pair (x, y) ∈ R, then x is
related to y under the relation. We write this as x ' y. A function
is a special sort of relation which has the extra requirement that
elements of A to have only one corresponding element of B.
Examples: 1) the relation of knowing each other: S is the set of
all people in the world, and
R = {(a, b) ∈ S × S : a knows b}.
2) greater than or equals: S = R and
R = {(a, b) ∈ R × R : a ≥ b}
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Equivalence relations
An equivalence relation is a relation that has the three key properties of equality: reflexiveness (a = a), symmetry (if a = b then
b = a), and transitivity (if a = b and b = c, then a = c). These
are the properties you use all the time in solving equations.
Formally, a relation R on a set A is an equivalence relation if:
• It is reflexive: for all a ∈ A, (a, a) ∈ R.
• It is symmetric: for all a, b ∈ A, if (a, b) ∈ R then (b, a) ∈ R.
• It is transitive: for all a, b, c ∈ A, if (a, b) ∈ R and (b, c) ∈ R
then (a, c) ∈ R.
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The relation an the real numbers “greater than or equal to” is
reflexive and transitive, but not symmetric.
The relation on the real numbers “is within a distance 1” is
reflexive and symmetric but not transitive.
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