Complement (set theory)

In set theory, a complement of a set A refers to things not in (that is, things outside of) A. The relative complement of A with respect to a set B is the set of elements in B but not in A. When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of all elements in U but not in A.

Relative complement

If A and B are sets, then the relative complement of A in B, also termed the set-theoretic difference of B and A, is the set of elements in B, but not in A.

The relative complement of A in B is denoted B ∖ A according to the ISO 31-11 standard (sometimes written B – A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b – a, where b is taken from B and a from A).

Formally

Examples:

If is the set of real numbers and is the set of rational numbers, then is the set of irrational numbers.

The following lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.

Difference (album)

Difference is the third album of the Finnish power metal band Dreamtale released on 25 April 2005 on Spinefarm Records. It was also released in Japan. It features on vocals for the first time in the band Jarkko Ahola of the band Teräsbetoni.

Track listing

Personnel

Subtraction

Subtraction is a mathematical operation that represents the operation of removing objects from a collection. It is signified by the minus sign (−). For example, in the picture on the right, there are 5 − 2 apples—meaning 5 apples with 2 taken away, which is a total of 3 apples. Therefore, 5 − 2 = 3. Besides counting fruits, subtraction can also represent combining other physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.

Subtraction follows several important patterns. It is anticommutative, meaning that changing the order changes the sign of the answer. It is not associative, meaning that when one subtracts more than two numbers, the order in which subtraction is performed matters. Subtraction of 0 does not change a number. Subtraction also obeys predictable rules concerning related operations such as addition and multiplication. All of these rules can be proven, starting with the subtraction of integers and generalizing up through the real numbers and beyond. General binary operations that continue these patterns are studied in abstract algebra.

Somebody

Somebody may refer to:

Albums

Songs

See also

Somebody (Mark Wills song)

"Somebody" is a country music song written by Dave Berg, Sam Tate, and Annie Tate. It was initially recorded by singer Mark Wills for his 2001 studio album Loving Every Minute. Reba McEntire later recorded the same song for her 2003 album Room to Breathe, releasing it as that album's second single in January 2004. By August of that year, McEntire's version had reached the top of the Billboard Hot Country Singles & Tracks (now Hot Country Songs) charts.

Chart performance

The song debuted at number 55 on the U.S. Billboard Hot Country Singles & Tracks for the week ending January 17, 2004.

Year-end charts

References

External links

Somebody (Aerosmith song)

"Somebody" is the B-side to Aerosmith's first single, "Dream On", from their 1973 debut album, Aerosmith. Written by lead singer Steven Tyler and his friend Steven Emspak and released in June 1973, its A-side peaked at number 59 nationally but hit big in the band's native Boston, where it was the number 1 single of the year on the less commercial top 40 station, WBZ-FM, number 5 for the year on highly rated Top 40 WRKO-AM, and number 16 on heritage Top 40 WMEX-AM.

"Somebody" is driven by a basic blues guitar riff and Tyler's lyrics tell the story of a character trying to search for the woman of his dreams. The character is in need of someone who shares the work with him and has been through the same difficulties in their life as he has. Originally written in 1970.

Single track listings

7" 45 RPM

Live performances

Aerosmith regularly performed this song throughout the early seventies, the first known time was in Mendon, Massachusetts on November 6, 1970 at Nipmuc Regional High School. The last-known performance of the song was in Yokohama on June 26, 1988 at the Yokohama Cultural Gymnasium during the Permanent Vacation Tour.