Sets and Number Sets Notes

Sets and Basic Notions

• Set: a collection of objects
  • A member of a set is something that belongs to it; we write using the membership symbol: $x \in S$
• Subset: a set A is a subset of B if every element of A is also an element of B
  • Notation: $A \subseteq B$
  • Example (from transcript):
  • Let $A = {a, e, i, o, u}$ and $B = {a, e}$
  • Then $B \subseteq A$ is true
  • And $A \subseteq B$ is false
• Union: the union of A and B is the set of elements that are in A or in B or in both
  • Notation: $A \cup B$
  • Example: with the same A and B above
  • $A \cup B = {a, e, i, o, u}$
• Intersection: the intersection of A and B is the set of elements that are in both A and B
  • Notation: $A \cap B$
  • Example: with the same A and B above
  • $A \cap B = {a, e}$
• Common terminology
  • "Common" elements are those in the intersection
• Empty set: the set that contains no elements
  • Notation: $\emptyset$
  • Eg. the empty set has no members by definition
• Quick recap of the notations
  • Subset: $A \subseteq B$
  • Union: $A \cup B$
  • Intersection: $A \cap B$
  • Empty set: $\emptyset$

Sets of Real Numbers and Number Classes

• Sets of numbers discussed in transcript
  • Natural numbers
  • Whole numbers
  • Integers
  • Real numbers (sets commonly studied together with these)

Natural Numbers

• Symbol: $\mathbb{N}$
• Definition (as given): $\mathbb{N} = {1, 2, 3, 4, \dots}$
• Note on zero
  • Some conventions include 0 in the natural numbers; in others, 0 is in the whole numbers instead

Whole Numbers

• Often denoted by a symbol like $\mathbb{W}$ or sometimes $\mathbb{N}_0$
• Definition (as given): $\mathbb{W} = {0, 1, 2, 3, \dots}$
• Relationship to other sets
  • $\mathbb{N} \subseteq \mathbb{W}$ (if 0 is included in N in that convention, otherwise N is a subset of W depending on the definition used)

Integers

• Symbol: $\mathbb{Z}$
• Definition (as given): $\mathbb{Z} = {\dots, -2, -1, 0, 1, 2, \dots}$
• Relationships
  • $\mathbb{W} \subseteq \mathbb{Z}$
  • $\mathbb{N} \subseteq \mathbb{Z}$
  • Generally, all Natural and Whole numbers are integers

Real Numbers (brief context)

• Real numbers include all rational and irrational numbers; they form the continuum
• In lecture context, the above number sets (N, W, Z) are subsets of the real numbers: $\mathbb{N} \subseteq \mathbb{W} \subseteq \mathbb{Z} \subseteq \mathbb{R}$
• Notation for these standard sets: $\mathbb{R}$ for real numbers

Quick Examples and Connections

• Example recap (from transcript):
  • A = ${a, e, i, o, u}$
  • B = ${a, e}$
  • Subset checks:
  • $B \subseteq A$ is true (since every element of B is in A)
  • $A \subseteq B$ is false (A has elements not in B)
  • Unions and intersections using these sets:
  • $A \cup B = {a, e, i, o, u}$
  • $A \cap B = {a, e}$
• Numerical examples (from transcript):
  • Natural numbers: ${1, 2, 3, 4, \dots}$
  • Whole numbers: ${0, 1, 2, 3, \dots}$
  • Integers: ${\dots, -2, -1, 0, 1, 2, \dots}$
• Practical implications
  • These sets provide the foundation for counting, ordering, and measuring in math and applied disciplines
  • They establish basic building blocks for more advanced topics (e.g., number theory, algebra, analysis)

Summary of Key Points

• A set is a collection of objects; membership is denoted by $x \in S$
• A is a subset of B if every element of A is in B: $A \subseteq B$
• Union and intersection operators: $A \cup B$, $A \cap B$
• Empty set: $\emptyset$, a set with no elements
• Specific number sets:
  • $\mathbb{N} = {1, 2, 3, 4, \dots}$
  • $\mathbb{W} = {0, 1, 2, 3, \dots}$
  • $\mathbb{Z} = {\dots, -2, -1, 0, 1, 2, \dots}$
  • $\mathbb{R}$ contains all real numbers; standard hierarchy $\mathbb{N} \subseteq \mathbb{W} \subseteq \mathbb{Z} \subseteq \mathbb{R}$