Uniqueness quantification

In mathematics and logic, the phrase "there is one and only one" is used to indicate that exactly one object with a certain property exists. In mathematical logic, this sort of quantification is known as uniqueness quantification or unique existential quantification.

Uniqueness quantification is often denoted with the symbols "∃!" or ∃₌₁". For example, the formal statement

may be read aloud as "there is exactly one natural number n such that n - 2 = 4".

Proving uniqueness

The most common technique to proving uniqueness is to first prove existence of entity with the desired condition; then, to assume there exist two entities (say, a and b) that should both satisfy the condition, and logically deduce their equality, i.e. a = b.

As a simple high school example, to show x + 2 = 5 has only one solution, we assume there are two solutions first, namely, a and b, satisfying x + 2 = 5. Thus

By transitivity of equality,

By cancellation,

This simple example shows how a proof of uniqueness is done, the end result being the equality of the two quantities that satisfy the condition. However, that existence/expressibility must be proven before uniqueness, or else we cannot even assume the existence of those two quantities to begin with our current knowledge to date.

Unique (DJ Encore album)

Unique is the second album by the Danish eurodance producer DJ Encore, with vocals by Swedish singer Johanna Elkesdotter, whereas the first album featured the vocals of Danish singer Engelina. It was released on May 8, 2007 on US label Koch Records.

The song "You Can Walk on Water" is a cover version of the 2002 song "Walk on Water (Baby U Can)" by Danish dance act Catch. "Out There" was originally released on DJ Encore's first album, Intuition, with vocals by Engelina.

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Unique (band)

Unique was a short-lived post-disco studio act from New York, best known for their crossover number "What I Got Is What You Need" released in 1983 for a well-established dance label, Prelude Records. The group was formed by producer/songwriter Deems J. Smith in 1982 and consists of Smith and Darryl K. Henry. The said hit song somewhat established itself on the Billboard Dance Singles and Black Singles charts and even scored over the atlantic reaching No. 27 on UK Singles Chart.

History

Around 1982, record producer Deems J. Smith hired studio musicians to appear in a project he named Unique. He was signed to dance record label Prelude and under this name released two singles: "What I Got Is What You Need" in 1983 and "You Make Me Feel So Good" in 1984. Other people involved in the group were Mona Maria Norris and Darryl K. Henry who co-wrote "What I Got Is What You Need." Smith wrote the second song alone and it was mixed by an aspiring dance-pop producer Shep Pettibone.

Chart performance

Morning

Morning is the period of time between midnight and noon or, more commonly, the interval between sunrise and noon. Morning precedes afternoon, evening, and night in the sequence of a day. Originally, the term referred to sunrise.

Etymology

The name (which comes from the Middle English word morwening) was formed from the analogy of evening using the word "morn" (in Middle English morwen), and originally meant the coming of the sunrise as evening meant the beginning of the close of the day. The Middle English morwen dropped over time and became morwe, then eventually morrow, which properly means "morning", but was soon used to refer to the following day (i.e., "tomorrow"), as in other Germanic languages—English is unique in restricting the word to the newer usage. The Spanish word "mañana" has two meanings in English: "morning," and "tomorrow," along with the word "morgen" in Dutch and German which also means both "morning," and "tomorrow." Max Weber, (General Economic History, pp23) states that the English word "morning" and the German word "Morgen" both signify the size of land strip "which an ox could plow in a day without giving out". "Tagwerk" in German, and "a day's work" in English mean the same. A Good morning in this sense might mean a good day's plow.

Morning (Clare Fischer composition)

"Morning" is a Latin Jazz standard written by American pianist/composer/arranger Clare Fischer, first heard on his 1965 LP, Manteca!, Fischer's first recording conceived entirely in the Afro-Cuban idiom, which, along with the Brazilian music he had explored at length over the previous three years, would provide fertile ground for Fischer's musical explorations over the next half-century.

Form

"Morning" was Fischer's first - and, to this day, his most famous - contribution to the then recently evolved cha-cha-chá genre. Its structure is the standard A-A-B-A, 32 measures in length. In practice, however, the song's debut recording does take one significant detour, paying unashamed homage to one of its composer's primary musical influences in the process, when, halfway through trombonist Gil Falco's solo, instead of proceeding to the bridge, "Morning" morphs into a 16-bar development of the principal 2-measure motif of "Spring Rounds," the fourth section from Stravinsky's The Rite of Spring.

Morning (disambiguation)

The morning is the part of the day from either midnight or dawn to noon.

Morning may also refer to:

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See also