Which statement is correct for (ℚ,ast) with aast b=a+b+1?

It is a group, with identity -1 and inverse of a equal to -2-a

It is not a group because ast is not associative

It is a group, with identity 0 and inverse of a equal to -a

It is not a group because it is not closed

Exercise · ZW-GROUP-011 · ★★★ Advanced

Which statement is correct for \((\mathbb{Q},\ast)\) with \(a\ast b=a+b+1\)?

Choose one option.

Show answer

D. It is a group, with identity \(-1\) and inverse of \(a\) equal to \(-2-a\)

Explanation

The identity \(e\) must satisfy \(a\ast e=a\), so \(e=-1\). The inverse of \(a\) satisfies \(a\ast b=-1\), i.e. \(b=-2-a\). Associativity holds: both sides give \(a+b+c+2\).

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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.

\[(G,\ast)\]

Which statement is correct for \((\mathbb{Q},\ast)\) with \(a\ast b=a+b+1\)?

Explanation

Review the concept: Group

More practice problems: group