Algebra: Sets and Operations with Sets

Sets. Ways in which a set can be defined: listing its elements or selecting a property common to all its elements. Venn (Euler) diagrams.

• elements of a set
• subsets
• universal or reference set
• complement or negation
• empty or null set
• disjoint sets

Operations with sets

• union (either/or): cup
• intersection (both/and): cusp
• negation/complement (but; everything except)
• implies
• logical equivalence

Laws of Operations with Sets

• Symmetrical difference or the disjunctive sum (exclusive or)
• DeMorgan's Laws (see Ayres, page 5).

Musical examples

• Complementarity of chords and scales (also see Anatol Vieru - :The Book of Modes)
  • Debussy - Voiles (piano prelude)
  • Luigi Nono - Guernica
  • Aurel Stroe - Piano Sonata
  • Skriabin - Prometheus, scales and quartal chord
  • combinatoriality: Schoenberg - Variations for Orchestra, Theme.
• Complex structures as the result of operatins with sets
  • Richard Strauss - Elektra: whole-tone fragment in Elektra's monologue (act 1).
  • Messiaen's modes with limited transpositions
  • Xenakis - Herma

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