set

Examples of set in the following topics:

• Setting Goals

  • Locke began to examine goal setting in the mid-1960s and continued researching goal setting for thirty years.
  • Locke derived the idea for goal-setting from Aristotle's form of final causality.
  • Goal setting and feedback go hand in hand, for without feedback, goal setting is unlikely to work.
  • Athletes set goals during the training process.
  • Goal-setting is used in business for sustainability, progress, and continued success.

• Set Class and Prime Form (2)

  • Analytically, the concept of set class is useful because it can show coherence in a composition.
  • Set class lists reveals all of the other possibilities.
  • Most of these set-class lists are organized similarly.
  • Set classes that have the same number of notes in them (we say that they have the same "cardinality") are grouped together: trichords (three-note pitch-class sets) sit together, as do nonachords (nine-note pitch-class sets), and so on.
  • Prime form for each set class is show in parenthesis.

• Sets of Numbers

  • Sets are one of the most fundamental concepts in mathematics.
  • The second way of describing a set is through extension: listing each member of the set.
  • All set operations preserve this property.
  • A subset is a set whose every element is also contained in another set.
  • For example, if every member of set $A$ is also a member of set $B$, then $A$ is said to be a subset of $B$.

• Unions and Intersections

  • The union of two or more sets is the set that contains all the elements of each of the sets; an element is in the union if it belongs to at least one of the sets.
  • To find the union of two sets, list the elements that are in either (or both) sets.
  • The intersection of two or more sets is the set of elements that are common to each of the sets.
  • To find the intersection of two (or more) sets, include only those elements that are listed in both (or all) of the sets.
  • The shaded Venn Diagram shows the union of set $A$ (the circle on left) with set $B$ (the circle on the right).

• Defining regular equivalence

  • This is because the concept of regular equivalence, and the methods used to identify and describe regular equivalence sets correspond quite closely to the sociological concept of a "role."
  • That is, regular equivalence sets are composed of actors who have similar relations to members of other regular equivalence sets.
  • Susan and Deborah form a regular equivalence set because each has a tie to a member of the other set.
  • Inga and Sally form a set because each has a tie to a member of the other set.
  • In regular equivalence, we don't care which daughter goes with which mother; what is identified by regular equivalence is the presence of two sets (which we might label "mothers" and "daughters"), each defined by its relation to the other set.

• Interval Notation

  • Interval notation uses parentheses and brackets to describe sets of real numbers and their endpoints.
  • A "real interval" is a set of real numbers such that any number that lies between two numbers in the set is also included in the set.
  • Other examples of intervals include the set of all real numbers and the set of all negative real numbers.
  • The set of all real numbers is the only interval that is unbounded at both ends; the empty set (the set containing no elements) is bounded.
  • Use interval notation to show how a set of numbers is bounded

• Introduction to describing structural equivalence sets

  • One very useful approach is to apply cluster analysis to attempt to discern how many structural equivalence sets there are, and which actors fall within each set.
  • What the similarity matrix and cluster analysis do not tell us is what similarities make the actors in each set "the same" and which differences make the actors in one set "different" from the actors in another.
  • A very useful approach to understanding the bases of similarity and difference among sets of structurally equivalent actors is the block model, and a summary based on it called the image matrix.

• Complements

  • The literal complement of a pitch-class set is every pitch not included in that set.
  • When we put both of those pitch-class sets in prime form, the two are said to be abstract complement:
  • On the set-class list, abstract complements are listed next to one another, and they have a very interesting intervallic relationship, as you can see by comparing their IC vectors.
  • Complementary set classes have a similar "distribution" of intervals.
  • Below, you'll see that the set (012345678) has exactly 6 more of each type of interval class than does its complement (012).

• Set Class and Prime Form (1)

  • These groups of pitch classes are called pitch-class sets.
  • We've already seen sets of pitch-classes, though we haven't really been calling them that.
  • In order for a pitch-class set to be transpositionally or inversionally related to some other pitch class set, they must share the same collection of intervals.
  • All pitch-class sets that are transpositionally and inversionally related belong to the same set class, and they are represented by the same prime form.
  • We follow a simple process to put a pitch-class set in prime form:

• Averages

  • The arithmetic mean, or average, of a set of numbers indicates the "middle" or "typical" value of a data set.
  • This is simply a mathematical way of writing "the mean equals the sum of all of the values in the set, divided by the number of values in the set."
  • To see how this applies to an actual set of numbers, consider the following set: $\{3,5,10\}$.
  • Next, divide their sum by 3, the number of values in the set:
  • Therefore, the average of the set of numbers $\{3,5,10\}$ is 5.