Subset

Rate It! (0 ratings)
  • Years Active: 2000s

Albums

Albums (1) Compilations (1) Singles (3) Sort By: Most Downloaded

Biography All Music Guide Wikipedia

All Music Guide:

Affiliating noise rock inspirations and conceptual melodies, Subset's compositions promote new views for the alternative pop/rock music fields. Inspired by bands like Sonic Youth, the Texas-born trio explores its own methods within experimental rock marks. Assembling in 1999, Subset combined the efforts of Lindsey Simon (vocals, guitar, bass) and Nathan Fish (vocals, guitar, bass). Joel Fish, Nathan's brother, was the next addition to the lineup, at the time assuring the trio's keyboard section. Following a number of rehearsals, the Texans delivered their first single, Circuitis, receiving considerable praise. Joining the band shortly afterward, Tom Hudson finally filled in the crew's drum section, months before Joel parted ways with the band, therefore fixing the set as a three-piece. Still, Subset's shows continued to gain exciting reviews from the media, and an always rising fan base. Featured on a number of local compilation discs, the band entered the studio in late 1999 to record their first album. Overpass hit the record stores in 2000 as the trio's debut full-length on the Post-Parlo Records label.

Wikipedia:

Euler diagram showing is a proper subset of and conversely is a proper superset of

In mathematics, especially in set theory, a set is a subset of a set , or equivalently is a superset of , if is "contained" inside . and may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment.

Definitions

If and are sets and every element of is also an element of , then:

is a subset of (or is included in) , denoted by ,or equivalently is a superset of (or includes) , denoted by

If is a subset of , but is not equal to (i.e. there exists at least one element of B not contained in ), then

is also a proper (or strict) subset of ; this is written as or equivalently is a proper superset of ; this is written as

For any set , the inclusion relation ⊆ is a partial order on the set of all subsets of (the power set of ).

The symbols ⊂ and ⊃

Some authors use the symbols ⊂ and ⊃ to indicate "subset" and "superset" respectively, instead of the symbols ⊆ and ⊇, but with the same meaning. So for example, for these authors, it is true of every set that  ⊂ .

Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, in place of ⊊ and ⊋. This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if  ≤  then may be equal to , or maybe not, but if  < , then definitely does not equal , but is strictly less than . Similarly, using the "⊂ means proper subset" convention, if  ⊆ , then may or may not be equal to , but if  ⊂ , then is definitely not equal to .

Examples

The set {1, 2} is a proper subset of {1, 2, 3}.Any set is a subset of itself, but not a proper subset.The empty set { }, denoted by ∅, is also a subset of any given set . It is also always a proper subset of any set except itself.The set {: is a prime number greater than 2000} is a proper subset of {: is an odd number greater than 1000}The set of natural numbers is a proper subset of the set of rational numbers and the set of points in a line segment is a proper subset of the set of points in a line. These are examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole; such cases can tax intuition.

Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (, ) is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal is identified with the set [] of all ordinals less than or equal to , then ≤ if and only if [] ⊆ [].

For the power set of a set , the inclusion partial order is (up to an order isomorphism) the Cartesian product of = || (the cardinality of ) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating = {, , …, } and associating with each subset ⊆ (which is to say with each element of 2) the -tuple from {0,1} of which the th coordinate is 1 if and only if is a member of .
more »