Insiemi e Operazioni con Insiemi

Insiemi

• Sets are collections of well-defined objects.
• Elements are the objects within a set.
• Set notation uses curly braces {} to enclose elements.
• Examples: {1, 2, 3}, {a, b, c}, {x | x is a positive integer}
• Empty set (null set): the set with no elements, denoted as {} or Ø.
• Subsets: A set A is a subset of a set B if every element of A is also an element of B. (denoted as A ⊆ B)
• Proper subsets: A is a proper subset of B if A is a subset of B and A is not equal to B (denoted as A ⊂ B)
• Set equality: Two sets are equal if they have exactly the same elements.
• Set operations: union (∪), intersection (∩), difference (-), complement (').
• Union: A ∪ B = {x | x ∈ A or x ∈ B or both}
• Intersection: A ∩ B = {x | x ∈ A and x ∈ B}
• Difference: A - B = {x | x ∈ A and x ∉ B}
• Complement: The complement of A (with respect to a universal set U), denoted as A', is the set of all elements in U that are not in A. (A' = U - A)
• Cardinality: The number of elements in a set. Denotes as |A|
• Countable and uncountable sets: Countable sets are those that can be put into a one-to-one correspondence with the natural numbers (N).
• Venn diagrams: visual representations of sets and their relationships.

Numeri Complessi

• Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1).
• a is the real part and bi is the imaginary part.
• Real numbers are a special case of complex numbers where b = 0.
• Imaginary numbers are a special case of complex numbers where a = 0.
• Complex numbers can be represented geometrically on the complex plane.
• The real axis represents the real part, and the imaginary axis represents the imaginary part.
• Operations on complex numbers:
  • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
  • Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
  • Multiplication: (a + bi) * (c + di) = (ac - bd) + (ad + bc)i
  • Division: (a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)] = [(ac + bd) + (bc - ad)i] / (c² + d²)
• Conjugate of a complex number z = a + bi is z* = a - bi
• Modulus or absolute value of a complex number z = a + bi is |z| = √(a² + b²)
• Polar form of complex numbers: z = r(cos θ + i sin θ) where r = |z| and θ is the argument (angle) of z
• Euler's formula: e^(iθ) = cos θ + i sin θ
• De Moivre's theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ)
• Roots of complex numbers: Find the n-th roots of a complex number z.
• Geometric interpretation of complex number operations and properties.