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GLOSSARY
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Posted 5 June 2009
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each
The word each may generally be used in the same way as all, every and any to form a universal quantifier. However it is not common to see an assertion worded as “Each multiple of 4 is even”. “Each” is more commonly used before a noun that is the object of a preposition (expecially “for”) or a verb, to have the effect as a distributive plural.
Examples
· “For each even number n there is an integer k for
which n = 2k.”
· “A binary operation ∗ on a set is a rule that assigns to each ordered pair of
elements of the set some element of the set.”
· “Five students have two pencils each.” This means that each of the five students has two pencils (a different two for each student). This usage occurs in combinatorics, for example. Some students do not grasp the significance of a postposited “each” as in this sentence.
element
empty set
equals, equation
2 + 2 and 1 + 3 are two different names of the same math
object. The statement 2 + 2 =
1 + 3 (“2 + 2 equals 1 + 3”) means
just that. Such an expression is called
an equation. There is more
information in the Wikipedia article.
Equations and expressions
An equation may have an intent that goes beyond the bare statement that two things are the same.
2 + 2 and 1 + 3 are expressions that denote a number. They denote the same number, namely 4. But the expression “2 + 2” contains more information than naming the number 4. It expresses 4 as a particular sum.
This phenomenon
is a general fact about equations. It
is uninteresting to say an object is equal to itself. It may be more interesting to say that this expression names the same object as that expression. Because the expressions may contain additional
information about the object, an equation may communicate information about
the object. Just what information is being communicated
depends on the context, and may not be obvious.
Examples
¨
The intent of the equation for a grade school student may be a
“multiplication fact”: Multiply 2 by 3
gives 6. The new information is 6.
¨ When I was a child, I
discovered that , which I thought was
quite wonderful. The new information
there was that is a factorization of 111 – it was the 3 and the 37 that constituted new
information.
¨ An equation containing variables may be an
identity, meaning it is a statement that the two objects are the same for all
values of the variables that satisfy the hypotheses up to this point in the
text. More about this under identity.
¨ An equation containing variables may be a constraint.,
for example, “Now suppose ”. But constraints do not have to be equations.
¨ An equation containing variables may be the defining
equation for a function.
equipped
Used to associate the structure attached to a set to make up a mathematical structure. Also endowed.
Example
A semigroup is a set equipped with [endowed with] an associative binary operation.
equivalent
¨ Two assertions are equivalent (sometimes “logically equivalent”) if they necessarily have the same truth values.
¨ Two math objects may be equivalent by an equivalence relation either explicitly given or known from context.
establish notation
exists
evaluate
To evaluate a function f at an input x is to find the value of f(x).
every
.