🦭 Number Sets (ℕ, ℤ, ℚ, ℝ)
Learn the difference between natural numbers ℕ and integers ℤ. Practice classifying finite and infinite sets.
ℕ and ℤ — Natural & Integer Sets
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A set S is shown below. Choose the correct statement describing the relationship between S, ℕ, and ℤ.
Learning Topics
Natural Numbers ℕ and Integers ℤ
Natural Numbers (ℕ)
The natural numbers are the counting numbers, starting from 0:
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 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)
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)
Learning Topics
Frequently Asked Questions
What are Natural Numbers (ℕ)?
Natural numbers are the non-negative whole numbers: ℕ = {0, 1, 2, 3, …}. They start at 0 and continue infinitely in the positive direction.
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.
Does the set {−3, -1, 0, 2} belong to ℕ or ℤ?
It belongs to ℤ but NOT to ℕ, because it contains negative numbers (−3 and −1). ℕ only contains non-negative numbers.
What if a set contains decimals like {1.5, 2.5}?
Decimals and fractions are neither natural numbers nor integers. A set like {1.5, 2.5} does not belong to ℕ or ℤ. It belongs to larger number sets like the rationals (ℚ) or reals (ℝ).