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Home » Determine whether the following operation define a binary operation on the given set or not: (i) ‘*’ on N defined by a * b = ab for all a, b ∈ N. (ii) ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.
Determine whether the following operation define a binary operation on the given set or not: (i) ‘*’ on N defined by a * b = ab for all a, b ∈ N. (ii) ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.
Binary Operations | CBSE | Class 12 | Exercise 3.1 | Maths | RD Sharma
Determine whether the following operation define a binary operation on the given set or not: (i) ‘*’ on N defined by a * b = ab for all a, b ∈ N. (ii) ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.

(i) Given ‘*’ on N defined by a * b = ab for all a, b ∈ N.

Let a, b ∈ N. Then,

    \[\begin{array}{*{35}{l}} {{a}^{b~}}\in ~N~~~~~~\left[ \because ~{{a}^{b}}\ne 0~and~a,\text{ }b~is~positive~integer \right] \\ \Rightarrow ~a~*~b~\in ~N \\ \end{array}\]

Therefore,

a * b ∈ N, ∀ a, b ∈ N

Thus, * is a binary operation on N.

(ii) Given ‘O’ on Z defined by a O b = ab for all a, b ∈ Z.

Both a = 3 and b = -1 belong to Z.

    \[\begin{array}{*{35}{l}} \Rightarrow \text{ }a~*~b\text{ }=\text{ }{{3}^{-1}} \\ =\text{ }1/3\text{ }\notin \text{ }Z \\ \end{array}\]

Thus, * is not a binary operation on Z.

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