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A Brief Overview of Logic
The following theorems of logic are used frequently in mathematical writing:
|Short But Important Words |
||IF ..., THEN ....
||IF AND ONLY IF
|Quantification (how many such and such are we discussing?)
"For all" phrases: for every x, for any y, for all z, given any x
for whatever y, given an arbitrary x, unquantified ("free") variables
|| "There exists" phrases: for some x, for at least one y,
there is a z
||Unique means one and only one. In mathematics, the also signals unique. For example,
one writes "the additive identity" only after proving that there is one and only one additive identity. The articles a
or an indicate that there is no claim for uniqueness. For example, "a left-handed inverse" might
be unique (but not known to be unique), might be known to be unique (but that fact isn't of interest to the writer), or
might not be unique.
|Rules of Equality|
For all a, a=a (reflexive). If a=b, then b=a (symmetric).
If a=b and b=c, then a=c (transitive).
Generally speaking, equal objects may be substituted
for one another in various expressions. For example, if a=b, then cos(a)=cos(b) and
a 3=b3. Since Bill Clinton is the U.S. President and Bill Clinton is the husband of
Hillary, the U.S. President is the husband of Hillary.