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namely

Used to indicate that what follows is one of the following:

¨ A definition or more detailed description of what precedes.

¨ The particular example of what precedes that the author is going to talk about.

Example

“Let G be an Abelian group, namely a group whose multiplication is commutative.” What comes after “namely” is simply the definition of “Abelian group”.

Example

“We now consider a specific group, namely ”. Here “ ” identifies the particular object alluded to by “a specific group”.

Example

“12 has two prime factors, namely 2 and 3.” The statement “12 has two prime factors” claims the existence of one or more objects, which are in fact 2 and 3.

Note

A statement such as “210 has several prime factors, namely 2 and 3” sounds incorrect to me, since 210 also has 5 and 7 as prime factors. I would use “for example” or “including” instead of “namely” here.

natural number

necessary

Q is necessary for P if P implies Q, in other words if the conditional assertion “If P, then Q” is true. Examples are given under conditional assertion.

The motivation for the word “necessary” is that the assertion “P implies Q” is logically equivalent to “not Q implies not P” (see contrapositive), so that for P to be true it is necessary in the usual sense of the word for Q to be true.

not

See negation.

nonnegative

A real number r is nonnegative if it is not negative, in other words if .

nonincreasing

notation

Notation is a system of signs and symbols used as a representation of something. The word notation may be used to refer to a whole system or to a way of representing a particular object.

Examples

¨ The symbolic language of mathematics is a system of notation.

¨ The standard notation for music is a system of notation.

¨ The usual notation for the empty set is “ ”, although some writers misunderstand it and use the Greek letter (phi).

¨ The notation {1, 2, 3} refers to the set whose elements are exactly 1, 2 and 3.

Remarks

¨ It would likely be better to say “symbol” instead of “notation” when the notation is a single symbol, such as .

¨ The notation for a math object, if it is simple enough, may be a particular example of an image we associate with the object. More here.

¨ The same math object may be referred to by many different notations (more here).

Establish notation

Mathematicians may say, “Let's establish some notation”, meaning they will introduce a specific combination of certain symbols to refer to a particular mathematical object. This is a type of definition on the fly, so to speak. See the first paragraph of this paper for a good example. See also fix and let.