Show that $⋆$ on $Z^{+}$ defined by $a ⋆ b = | a - b |$ is not a binary operation.

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Solution

A binary operation on a set is an operation whose domains and the codomain are the same set.

Let a be an arbitrary positive integer. Then,
$a ⋆ a = | a - a | = 0$ , which is not a positive integer.
$∴ ⋆$ on $Z^{+}$ is not a binary operation.

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Show that ⋆ on Z+ defined by a⋆b=|a−b| is not a binary operation.

A binary operation on a set is an operation whose domains and the codomain are the same set.Let a be an arbitrary positive integer. Then,a⋆a=|a−a|=0, which is not a positive integer.∴⋆ on Z+ is not a binary operation.

Show that $⋆$ on $Z^{+}$ defined by $a ⋆ b = | a - b |$ is not a binary operation.

A binary operation on a set is an operation whose domains and the codomain are the same set.

Let a be an arbitrary positive integer. Then,
$a ⋆ a = | a - a | = 0$ , which is not a positive integer.
$∴ ⋆$ on $Z^{+}$ is not a binary operation.