A Brief Overview of Logic

Short But Important Words
Conjunction Alternation Negation Conditional Biconditional
Quantification (how many such and such are we discussing?)
Universal Quantification: for every x, for any y, for all z, given any x for whatever y, given an arbitrary x, unquantified ("free") variables

Existential Quantification: for some x, for at least one y, there is a z

Unique means one and only one. In logic and 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.
The following theorems of logic are used frequently in mathematical writing: