In mathematics, an indexed family is a collection of values that are associated with indexes. For example, a family of real numbers, indexed by the integers is a collection of real numbers, where each integer is associated with one of the real numbers.

Formally, an indexed family is the same thing as a function. A function with domain J and codomain X is equivalent to a family of elements of X indexed by elements of J. The only difference is that indexed families are thought of as collections instead of as functions. A value is considered to be an element of a family whenever it is an element of the image of the family's underlying function.

When a function f : J → X is treated as a family, J is called the index set of the family, the functional image f(j) for j ∈ J is denoted x_j, and the mapping f is denoted {x_j}_j∈J or simply {x_j}.

Next, if the set X is the power set of a set U, then the family {x_j}_j∈J is called a family of sets indexed by J .

Definition. Let X and I be any sets. Then by family of elements in X indexed by I , we mean a function Failed to parse (Missing texvc executable; please see math/README to configure.): x : I \mapsto X . An indexed family is denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \{ x_i \}_{i \in I} , where it is understood that there is a function x that maps i to Failed to parse (Missing texvc executable; please see math/README to configure.): x_i := x(i) \, .

The Basics ( /ˈbæzɪtʃ/ BAZZ-ich) are a three-piece band from Melbourne, Australia, formed by Wally De Backer and Kris Schroeder in 2002. Their style has been described as anything from 'indie-pop' to 'rock'n'roll' to 'pop-rock', though their records show they span a wide range of genres, including reggae, ska, country, and electro-pop. They are "recognised as one of Australia's hardest-working bands".

With close to 1000 shows in their ten-year career, their live performances are well-known for their sense of humour and energy.

The Basics were formed after Kris met Wally at a party in Melbourne. The event was to see off the young aspiring producer, who was about to depart for Los Angeles, and together with a number of musician friends, the two jammed on some blues-rock standards. Later that night, they bonded over a mutual love of The Beatles, 70s and 80s cartoon theme songs and old Sierra adventure games.

Initially, they played around Melbourne as an acoustic guitar/drums combo, starting at The Opposition in Frankston and the House of Fools in Footscray, where in late 2002 they met Michael Hubbard and later invited him to join them on electric guitar. Though lacking any real experience with the instrument, Kris willingly purchased his first bass guitar, and the trio was born. During this time, the group start performing songs with more complicated three-part harmony.

Doug Wright (born 1962) is an American playwright, librettist, and screenwriter. He received the Pulitzer Prize for Drama in 2004 for his play, I Am My Own Wife.

Wright was born in Dallas, Texas. He attended and graduated from Highland Park High School, in a suburb of Dallas, Texas, where he excelled in the theater department and was President of the Thespian Club in 1981. He earned his bachelor's degree from Yale University in 1985. He earned his M.F.A. from New York University. He is a member of the Dramatists Guild and serves on the boards of Yaddo and New York Theatre Workshop. He is a recipient of the William L. Bradley Fellowship at Yale University, the Charles MacArthur Fellowship at the Eugene O'Neill Theater Center, an HBO Fellowship in playwriting and the Alfred Hodder Fellowship at Princeton University.

Wright’s play Quills premiered at Washington, D.C.'s Woolly Mammoth Theatre Company in 1995 and subsequently had its debut Off-Broadway at New York Theatre Workshop. The play recounts the imagined final days in the life of the Marquis de Sade. Quills garnered the 1995 Kesselring Prize for Best New American Play from the National Arts Club and, for Wright, a 1996 Village Voice Obie Award for Outstanding Achievement in Playwriting. In 2000, Wright wrote the screenplay for the film version of Quills which starred Geoffrey Rush.