Lesson 1

Sets of Numbers

  • Natural Numbers (N):
    • Numbers used for counting: N=1,2,3,4,…{N = {1, 2, 3, 4, …}}.
    • Do not include zero.
  • Whole Numbers:
    • Includes natural numbers and zero.
    • 0,1,2,3,…{ {0, 1, 2, 3,…}}
  • Integers (Z):
    • Includes natural numbers, zero, and the negatives of natural numbers.
    • Z=0,+1,−1,+2,−2,…=−2,−1,0,1,2,…{Z= {0, +1, -1, +2, -2, …} = {-2, -1, 0, 1, 2,…}}.
  • Rational Numbers (Q):
    • Numbers that can be expressed as a fraction of two integers, where the denominator is not zero.
    • Q=ab∣a,b∈Z,b≠0{Q = { \frac{a}{b} \mid a, b \in Z, b \neq 0 }}
    • Expressed in set builder notation.
  • Irrational Numbers:
    • Numbers that cannot be written as a quotient of integers (e.g., ππ, 2\sqrt{2}, 19\sqrt{19}).
    • No standard symbol to designate them.
  • Real Numbers (R):
    • Combination of rational and irrational numbers.
    • R=Q∪Qˉ{R = Q \cup \bar{Q}}, where Qˉ\bar{Q} represents the set of irrational numbers.

Relationships Between Number Sets

  • Subset Relationships:
    • Q⊂RQ \subset R (Rational numbers are a subset of real numbers).
    • N⊂ZN \subset Z (Natural numbers are a subset of integers).
    • N⊂QN \subset Q (Natural numbers are a subset of rational numbers).
    • Z⊂RZ \subset R (Integers are a subset of real numbers).

Closure Property

  • Definition:
    • A set is closed under an operation if performing that operation on members of the set always yields a member of the same set.
    • If the result is not a member of the same set, even once, the set is not closed under that operation.
  • Examples:
    • Natural numbers (N) are closed under addition and multiplication.
    • Natural numbers (N) are not closed under subtraction or division.
    • Integers (Z) are closed under addition, subtraction, and multiplication but not division.
    • Rational numbers (Q) are closed under addition, subtraction, multiplication, and division (excluding division by zero).
  • Illustrative Examples:
    • 5∈N5 \in N (5 is a member of the set of natural numbers).
    • −5∉N-5 \notin N (-5 is not a member of the set of natural numbers).
    • 111∉N\frac{1}{11} \notin N (1/11 is not a member of the set of natural numbers).

Key Concepts

  • Real numbers are the rational numbers together with the irrational numbers; designated with R.
  • Rational numbers are numbers of the form ab\frac{a}{b}, where a,b∈Z,b≠0a, b \in Z, b \neq 0 and designated with Q.
  • Integers are numbers  0,+1,−1,+2,−2,…\ {0, +1, -1, +2, -2, …} and designated with Z.