Exercise · ZW-GROUP-007 · ★★☆ Intermediate
\(\mathbb{N}=\{0,1,2,\ldots\}\) with the usual addition is not a group because...
Choose one option.
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- D. Not every natural has an additive inverse in ℕ
Explanation
With \(0\) in the set, \(\mathbb{N}\) is closed under addition, addition is associative and \(0\) is the identity. What fails is that, for example, \(3\) has no natural \(n\) with \(3+n=0\).
Review the concept: Group
A group is a nonempty set \(G\) together with an associative binary operation, with an identity element and an inverse for every element.
\[(G,\ast)\]
More practice problems: group
- ZW-GROUP-001 Which list contains the axioms of a group?
- ZW-GROUP-002 Which structure is a group?
- ZW-GROUP-003 Why is (ℤ, ·) not a group?
- ZW-GROUP-004 What is the identity of (ℤ, +)?
- ZW-GROUP-005 What is the inverse of 3 in (ℤ, +)?
- ZW-GROUP-006 Which structure is a group with the indicated operation?
- ZW-GROUP-008 The identity of a group must...
- ZW-GROUP-009 Which of these conditions is NOT a group axiom?