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Is ℕ ~ ℤ? (Naturals Equivalent to Integers)

publish date: 2026/05/23 21:45:36.902111 UTC

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Are the natural numbers \(\mathbb{N} = \{1,2,3,4,\ldots\}\) and the integers \(\mathbb{Z} = \{\ldots,-2,-1,0,1,2,\ldots\}\) equivalent sets?

No — Z has more elements since it includes 0 and negative numbers

Yes — a bijection f: Z → N exists, so they have the same cardinality ℵ₀

No — N is countable but Z is uncountable

Yes — but only because both are subsets of ℝ

Correct Answer

Yes — a bijection f: Z → N exists, so they have the same cardinality ℵ₀

Explanation

Yes. A bijection \(f: \mathbb{Z} \to \mathbb{N}\) can be defined: \(f(n) = 2n\) for \(n > 0\) and \(f(n) = -2n + 1\) for \(n \le 0\). This pairs: \(0 \leftrightarrow 1, 1 \leftrightarrow 2, -1 \leftrightarrow 3, 2 \leftrightarrow 4, -2 \leftrightarrow 5, \ldots\) So \(\mathbb{N} \sim \mathbb{Z}\) — both have cardinality \(\aleph_0\).

Reference

Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012

Quizzes you can take where this question appears

Cardinality