Number Sets (ℕ, ℤ, ℚ, ℝ): ℕ and ℤ — Natural & Integer Sets
Learn the difference between natural numbers ℕ and integers ℤ. Practice classifying finite and infinite sets.
ℕ and ℤ — Natural & Integer Sets
Natural Numbers ℕ and Integers ℤ
The natural numbers are the counting numbers, starting from 0:
ℕ = {0, 1, 2, 3, 4, …}They go on forever in the positive direction. Every natural number is also an integer.
Integers (ℤ)
The integers include all whole numbers — both positive and negative — and zero:
ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}ℤ extends infinitely in both directions. If a number has a negative sign but is still whole, it belongs to ℤ but not to ℕ.
What about fractions or decimals?
Numbers like 1.5, ½, or √2 belong to neither ℕ nor ℤ. They require larger sets like ℚ (rationals) or ℝ (reals).
Subset Notation:
• ⊆ means "subset" (e.g., S ⊆ ℕ means all elements of S are natural numbers).
• ⊄ means "not a subset" (e.g., S ⊄ ℕ means at least one element of S is not a natural number).
Relationship: ℕ ⊂ ℤ — every natural number is also an integer, but not every integer is natural.
Examples:
• {0, 1, 2, 3} ⊆ ℕ ✓ (all non-negative whole numbers)
• {−2, 0, 1, 5} ⊆ ℤ but ⊄ ℕ (contains a negative)
• {1.5, 2.5} ⊄ ℤ (decimals — neither ℕ nor ℤ)
• {0, 1, 2, …} = ℕ itself (infinite natural number set)
Frequently Asked Questions
What is a set and what are its elements?
A set is a collection of distinct objects or numbers called elements. We write a set by listing its elements inside curly braces, like A = {1, 2, 3}.
What do the symbols ∈ and ∉ mean?
The symbol ∈ means “belongs to” or “is an element of” (e.g. 2 ∈ {1, 2, 3} is true). The symbol ∉ means “does not belong to” (e.g. 4 ∉ {1, 2, 3} is true).
What is the union of two sets?
The union A ∪ B contains all elements that appear in A, in B, or in both. No duplicates are listed. Example: {1, 2} ∪ {2, 3} = {1, 2, 3}.
What is the intersection of two sets?
The intersection A ∩ B contains only elements that appear in both A and B. Example: {1, 2, 3} ∩ {2, 3, 4} = {2, 3}.
What are Integers (ℤ)?
Integers include all whole numbers — positive, negative, and zero: ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}. Every natural number is also an integer, but not every integer is natural.
What is the difference between real and complex numbers?
Real numbers include all rational and irrational numbers. Complex numbers include all real numbers as well as numbers containing the imaginary unit $i$ (where $i^2 = -1$), allowing us to solve roots of negative numbers.
Is zero positive or negative?
Zero is neither positive nor negative. It is the boundary between them.