Notes by David Groisser, Copyright c
±
1998
Mathematical grammar
and correct use of terminology
Sentences in mathematical writing often use mathematical symbols. These symbols
have very precise meanings. Some examples are
1. “=” stands for “equals”, “which equals”, or “is equal to”. It does not stand for
“Doing the next step in this problem, I arrive at the expression to the right of the
equals sign.”
2. “
∀
” stands for “for all”, “for every”, or “for each”.
3. “
∃
” stands for “there exists” or “there exist”.
4. “
⇒
” stands for “implies”, “implying” or “which implies”.
5. “
⇐
” stands for “which is implied by” (this can also be read “implies”, if you read
from right to left or from the bottom of a page up).
6. “
⇐⇒
” stands for “if and only if” or “which is equivalent to”.
7. “
⊂
” stands for “subset”, “in”, “is a subset of” (as in the sentence “
W
⊂
V.
”), “a
subset of”, or “be a subset of” (as in the sentence “Let
W
⊂
V.
”). A common
convention among mathematicians is that in a sentence, “subset” can be also used
as a preposition (a word like
in
or
on
that indicates the relation of two objects to
one another). Example
: In “Let
W
⊂
V
be a subspace”, “
⊂
” is a preposition,
and the sentence is read “Let
W
subset
V
be a subspace” or “Let
W
in
V
be a
subspace.” If we want the sentence instead to read “Let
W
, a subset of
V
, be a
subspace,” we have to punctuate it accordingly: “Let
W,
⊂
V
, be a subspace.” Using
mathematical symbols does not relieve the writer of the responsibility to punctuate
his or her sentences correctly; the symbols do not incorporate punctuation marks.
Your written work should have the property that, when the conventional English
meanings of your symbols are substituted for the symbols themselves, the result is a
collection of sentences with correct grammar and punctuation, with logical connections
between the sentences. In particular this applies to equations, which are examples of
sentences, and to strings of equations, which are often used as long sentences that detail
the logical ﬂow of an argument. A common way to achieve mathematical gibberish is
simply to write equations down on a page, with no words connecting them to indicate