In English, for a thing to be ``equal'' to another is to assert that two
*different* things are in fact the *same* - a rather oxymoronic
thing if one thinks about it. In mathematics it means that two
(possibly) different *symbols* in fact stand for the *same
thing* - a rather more sensible quality.

The simplest statement of equality, one that is *always* correct for
*any* possible symbol that stands anything at all is the *tautology*:

(3.1) |

That is, whatever stands for is the same as itself. It is what it
is. As a symbol for a number, it has whatever *value* it has. This
law is so fundamental that it is difficult to imagine any sort of
logical system where it is not true.

Simple or not, obvious or not, there are (amazingly enough) times we
will want to start with this in a derivation. Again observe that in
algebra the statement:

(3.2) |

While we are discussing basic equality, we will add to these fairly
obvious statements the less obvious rule of *transitivity*. If

(3.3) |

(3.4) |

(3.5) |