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# If <math>L\!</math> is a sign relation such that <math>F\!</math> is its DER on <math>S = I,\!</math> then the denotative equation <math>x ~\overset{L}{=}~ y</math> means that <math>(x, y) \in F,</math> in other words, that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math>

# If <math>L\!</math> is a sign relation such that <math>F\!</math> is its DER on <math>S = I,\!</math> then the denotative equation <math>x ~\overset{L}{=}~ y</math> means that <math>(x, y) \in F,</math> in other words, that <math>\operatorname{Den}(L, x) = \operatorname{Den}(L, y).</math>

The use of square brackets for denoting equivalence classes is recalled and extended in the following ways:

The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:

1. If E is an arbitrary equivalence relation,

# If <math>E\!</math> is an arbitrary equivalence relation, then <math>[x]_E\!</math> is the equivalence class of <math>x\!</math> under <math>E.\!</math>

then "[x]E" denotes the equivalence class of x under E.

# If <math>L\!</math> is a sign relation such that <math>\operatorname{Con}(L)</math> is a SER on <math>S = I,\!</math> then <math>[x]_L\!</math> is the SEC of <math>x\!</math> under <math>\operatorname{Con}(L).</math>

 

# If <math>L\!</math> is a sign relation such that <math>\operatorname{Der}(L)</math> is a DER on <math>S = I,\!</math> then <math>[x]^L\!</math> is the DEC of <math>x\!</math> under <math>\operatorname{Der}(L).</math>

2. If R is a sign relation such that Con(R) is a SER on S = I,

then "[x]R" denotes the SEC of x under Con(R).

 

3. If R is a sign relation such that Der(R) is a DER on S = I,

then "[x]R" denotes the DEC of x under Der(R).

By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.

By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.