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.

Track listing

References

External links

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

Toy

A toy is an item that can be used for play. Toys are generally played with by children and pets. Playing with toys is an enjoyable means of training young children for life in society. Different materials are used to make toys enjoyable to all ages. Many items are designed to serve as toys, but goods produced for other purposes can also be used. For instance, a small child may pick up a household item and "fly" it through the air as to pretend that it is an airplane. Another consideration is interactive digital entertainment. Some toys are produced primarily as collector's items and are intended for display only.

The origin of toys is prehistoric; dolls representing infants, animals, and soldiers, as well as representations of tools used by adults are readily found at archaeological sites. The origin of the word "toy" is unknown, but it is believed that it was first used in the 14th century. Toys are mainly made for children.

Playing with toys is important when it comes to growing up and learning about the world around us. Younger children use toys to discover their identity, help their bodies grow strong, learn cause and effect, explore relationships, and practice skills they will need as adults. Adults use toys to form and strengthen social bonds, teach, remember and reinforce lessons from their youth, discover their identity, exercise their minds and bodies, explore relationships, practice skills, and decorate their living spaces.

Walls Have Eyes

Walls Have Eyes is the fourth solo album released by British singer-songwriter Robin Gibb. It was released in November 1985 on EMI America Records in the US and Polydor Records throughout the rest of the world, and produced by Maurice Gibb and Tom Dowd. The two singles from the album, "Like a Fool" and "Toys", did not chart in the US and UK. Gibb did not release a solo album in eighteen years until 2003 with Magnet. Unlike Secret Agent that contains dance numbers, this album is containing mostly ballads.

Background

Writing credits

Barry Gibb co-wrote eight of the ten songs, but he was in the studio only to contribute a part lead vocal to "Toys". The credits for the songs are precisely stated as R. B. & M. Gibb in most cases, and B. R. & M. Gibb in others. Although the songs all have Robin's signature simplicity of form, Barry's hand is evident in the improved melody lines, especially in the verses.

Recording

Like its previous album Secret Agent it was recorded in Criteria Studios rather than the Middle Ear Studio which was owned by the Bee Gees.The only regulars from previous Gibb recordings was George Terry on guitar with Steve Farris of Mr. Mister. Session musician Phil Chen was credited as the bass player on the song "Gone with the Wind".

Toys (video game)

Toys is an action video game for the Super NES and Sega Genesis released in 1993.

The game is based on the 1992 film Toys starring Robin Williams. Chaos has been spread at a toy factory that must be stopped by the player.

Plot

An irresponsible young man (Leslie Zevo) wouldn't take over the company and now his father is dying. In order to get his young adult son to accept his new responsibilities, the father must force him to reclaim his toy factory from a straight-laced Army general (Lt. General Leland) that he has appointed as part of a "test of maturity."

Gameplay

Players are involved in a "toy war" between an army of military-style toys fought with a personalized action figure army using toys found in certain places in the game. The player will start each level only possessing a limited supply of toys, whereas the general will have a virtually unlimited supply of toys to throw at the player. These toys range from the realistic (bowling balls) to the cartoonish (i.e., peanut gun projectiles, radio-controlled cars, water balloons). Crashing a toy plane into General Zevo's windows allows players to beat the game. However, they have to navigate the plane through a scale model of Manhattan and avoid running out of power.