Phrase

In everyday speech, a phrase may be any group of words, often carrying a special idiomatic meaning; in this sense it is roughly synonymous with expression. In linguistic analysis, a phrase is a group of words (or possibly a single word) that functions as a constituent in the syntax of a sentence—a single unit within a grammatical hierarchy. A phrase appears within a clause, although it is also possible for a phrase to be a clause or to contain a clause within it.

Common and technical use

There is a difference between the common use of the term phrase and its technical use in linguistics. In common usage, a phrase is usually a group of words with some special idiomatic meaning or other significance, such as "all rights reserved", "economical with the truth", "kick the bucket", and the like. It may be a euphemism, a saying or proverb, a fixed expression, a figure of speech, etc.

In grammatical analysis, particularly in theories of syntax, a phrase is any group of words, or sometimes a single word, which plays a particular role within the grammatical structure of a sentence. It does not have to have any special meaning or significance, or even exist anywhere outside of the sentence being analyzed, but it must function there as a complete grammatical unit. For example, in the sentence Yesterday I saw an orange bird with a white neck, the words an orange bird with a white neck form what is called a noun phrase, or a determiner phrase in some theories, which functions as the object of the sentence.

Element (mathematics)

In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.

Sets

Writing A = {1, 2, 3, 4} means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example {1, 2}, are subsets of A.

Sets can themselves be elements. For example, consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.

The elements of a set can be anything. For example, C = { red, green, blue }, is the set whose elements are the colors red, green and blue.

Notation and terminology

The relation "is an element of", also called set membership, is denoted by the symbol "∈". Writing

means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A". The expressions "A includes x" and "A contains x" are also used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos strongly urged that "contains" be used for membership only and "includes" for the subset relation only.