math 101

Set

A collection of well-defined, distinct objects (elements) denoted by capital letters.

Finite Set

A set that has a countable number of elements.

Infinite Set

A set that has uncountable elements.

Empty Set

A set that contains no elements, denoted by ∅ or {}.

Union (A ∪ B)

Elements in either set A or set B.

Intersection (A ∩ B)

Elements common to both set A and set B.

Complement (A')

Elements not in set A but in the universal set.

Difference (A - B)

Elements in set A but not in set B.

Symmetric Difference

Elements in either set A or set B but not both.

De Morgan's Laws

The laws stating (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.

Natural Numbers (ℕ)

Counting numbers that include 1, 2, 3, and so on.

Integers (ℤ)

Numbers that include natural numbers, zero, and negatives.

Rational Numbers (ℚ)

Numbers expressible as a fraction a/b where a and b are integers and b ≠ 0.

Real Numbers (ℝ)

All numbers on the number line, including both rational and irrational numbers.

Complex Numbers (ℂ)

Numbers of the form z = x + iy, where x and y are real numbers.

Algebraic Form of Complex Numbers

A representation of complex numbers in the form z = x + iy.

Polar Form of Complex Numbers

A representation of complex numbers as z = r(cos θ + i sin θ).

De Moivre's Theorem

The theorem stating z^n = r^n (cos nθ + i sin nθ) for a complex number.

Reflexive Relation

A relation where every element is related to itself.

Symmetric Relation

A relation where if a is related to b, then b is related to a.

Transitive Relation

A relation where if a is related to b and b to c, then a is related to c.

Equivalence Relation

A relation that is reflexive, symmetric, and transitive.

Cartesian Product

The product A × B = {(a, b) | a ∈ A, b ∈ B}, forming the basis for relations.