In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as # to distinguish from the negation of equality (the denial inequality) ≠, which is weaker.

An apartness relation is a symmetric irreflexive binary relation with the additional condition that if two elements are apart, then any other element is apart from at least one of them (this last property is often called co-transitivity or comparison).

That is, a binary relation # is an apartness relation if it satifies:

The negation of an apartness relation is an equivalence relation, as the above three conditions become reflexivity, symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight. That is, # is a tight apartness relation if it additionally satisfies:

In classical mathematics, it also follows that every apartness relation is the negation of an equivalence relation, and the only tight apartness relation on a given set is the negation of equality. So in that domain, the concept is not useful. In constructive mathematics, however, this is not the case.

The Cure are an English alternative rock band formed in Crawley, West Sussex in 1976. The band has experienced several line-up changes, with frontman, vocalist, guitarist and principal songwriter Robert Smith being the only constant member. The Cure first began releasing music in the late 1970s with its debut album Three Imaginary Boys (1979); this, along with several early singles, placed the band as part of the post-punk and New Wave movements that had sprung up in the wake of the punk rock revolution in the United Kingdom. During the early 1980s, the band's increasingly dark and tormented music helped form the gothic rock genre.

After the release of Pornography (1982), the band's future was uncertain and Smith was keen to move past the gloomy reputation his band had acquired. With the 1982 single "Let's Go to Bed" Smith began to place a pop sensibility into the band's music (as well as a unique stage look). The Cure's popularity increased as the decade wore on, especially in the United States where the songs "Just Like Heaven", "Lovesong" and "Friday I'm in Love" entered the Billboard Hot 100 chart. By the start of the 1990s, The Cure were one of the most popular alternative rock bands in the world. The band is estimated to have sold 27 million albums as of 2004. The Cure have released thirteen studio albums, 10 EPs and over thirty singles during the course of their career. Since 2010, they have been working on a fourteenth studio album.

Papa Roach is an American rock band from Vacaville, California. Their first major-label release was the triple-platinum album Infest (2000). The group's success continued with their gold album Lovehatetragedy (2002), their platinum album Getting Away with Murder (2004), The Paramour Sessions (2006), and Metamorphosis (2009). The band's seventh album, Time for Annihilation, was released on August 31, 2010. Papa Roach has sold about 10 million album copies worldwide, and are known for their hits "Last Resort", "Between Angels And Insects", "She Loves Me Not", "...To Be Loved", "Scars", "Forever", and "Hollywood Whore".

The formation of Papa Roach began in January 1993, when lead singer Jacoby Shaddix and drummer Dave Buckner met on the Vacaville High School football field during an intense match, where the two ended up talking about music. They were later joined by trombonist Ben Luther and bass guitarist Will James. They decided to enter the school's talent show, where they performed a cover version of Jimi Hendrix's song "Fire", but they ultimately did not win the contest. In March 1993, trombonist Ben Luther was replaced by lead guitarist Jerry Horton from nearby Vanden High School. Horton was introduced to the band through his girlfriend, who was a fan. However, Jerry was at first apprehensive about joining the band because Jacoby, Buckner, and Will all went to different high schools. At this point, Papa Roach were practicing every day and touring on every gig they could get. Their first tour van was called Moby Dick, where Jacoby was inspired to come up with his first stage name "Coby Dick". The band's name comes from Shaddix's step-grandfather, Howard William Roatch, who was nicknamed Papa Roach. Roatch committed suicide in 2006 because he was diagnosed with terminal cancer. The band pay homage to him with The Paramour Sessions and during live performances of the song "Roses On My Grave".