Changes

The universe of discourse <math>X^\circ\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.

The universe of discourse <math>X^\circ\!</math> consists of two parts, a set of ''points'' and a set of ''propositions''.

: The points of <math>X^\circ</math> form the space:

The points of <math>X^\circ</math> form the space:

:: <math>X = \langle \mathcal{X} \rangle = \langle u, v \rangle = \{ (u, v) \} \cong \mathbb{B}^2.</math>

| <math>X \quad = \quad \langle \mathcal{X} \rangle \quad = \quad \langle u, v \rangle \quad = \quad \{ (u, v) \} \quad \cong \quad \mathbb{B}^2.</math>

: Each point in <math>X\!</math> may be described by means of a ''singular proposition'', that is, a proposition that describes it uniquely.  This form of representation leads to the following enumeration of points:

Each point in <math>X\!</math> may be described by means of a ''singular proposition'', that is, a proposition that describes it uniquely.  This form of representation leads to the following enumeration of points:

:: <math>X = \{\ (\!|u|\!)(\!|v|\!),\ (\!|u|\!) v,\ u (\!|v|\!),\ u v\ \} \cong \mathbb{B}^2.</math>

| '''<math>X \quad = \quad \{\!</math>&nbsp;<code>(u)(v)</code>&nbsp;,&nbsp;<code>(u)v</code>&nbsp;,&nbsp; <code>u(v)</code>&nbsp;,&nbsp;<code>uv</code>&nbsp;<math>\} \quad \cong \quad \mathbb{B}^2.</math>'''

: Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}</math> that the coordinate propositions <math>u\!</math> and <math>v\!</math> take on that point.  This form of representation leads to the following enumeration of points:

Each point in <math>X\!</math> may also be described by means of its ''coordinates'', that is, by the ordered pair of values in <math>\mathbb{B}</math> that the coordinate propositions <math>u\!</math> and <math>v\!</math> take on that point.  This form of representation leads to the following enumeration of points:

:: <math>X = \{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \} \cong \mathbb{B}^2.</math>

| <math>X \quad = \quad \{\ (0, 0),\ (0, 1),\ (1, 0),\ (1, 1)\ \} \quad \cong \quad \mathbb{B}^2.</math>

: The propositions of <math>X^\circ</math> form the space:

The propositions of <math>X^\circ</math> form the space:

:: <math>X^\uparrow = (X \to \mathbb{B}) = \{ f : X \to \mathbb{B} \} \cong (\mathbb{B}^2 \to \mathbb{B}).</math>

| <math>X^\uparrow \quad = \quad (X \to \mathbb{B}) \quad = \quad \{ f : X \to \mathbb{B} \} \quad \cong \quad (\mathbb{B}^2 \to \mathbb{B}).</math>

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.

As always, it is frequently convenient to omit a few of the finer markings of distinctions among isomorphic structures, so long as one is aware of their presence and knows when it is crucial to call upon them again.