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Home » Check the commutativity and associativity of each of the following binary operations: (i) ‘*’ on Z defined by a * b = a + b + a b for all a, b ∈ Z (ii) ‘*’ on N defined by a * b = 2ab for all a, b ∈ N
Check the commutativity and associativity of each of the following binary operations: (i) ‘*’ on Z defined by a * b = a + b + a b for all a, b ∈ Z (ii) ‘*’ on N defined by a * b = 2ab for all a, b ∈ N
Binary Operations | CBSE | Class 12 | Exercise 3.2 | Maths | RD Sharma
Check the commutativity and associativity of each of the following binary operations: (i) ‘*’ on Z defined by a * b = a + b + a b for all a, b ∈ Z (ii) ‘*’ on N defined by a * b = 2ab for all a, b ∈ N

(i)  to check: commutativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Z \\ =>\text{ }a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }+\text{ }b\text{ }+\text{ }ab \\ =\text{ }b\text{ }+\text{ }a\text{ }+\text{ }ba \\ =\text{ }b\text{ }*\text{ }a \\ =>a\text{ }*\text{ }b\text{ }=\text{ }b\text{ }*\text{ }a,\text{ }\forall \text{ }a,\text{ }b\text{ }\in \text{ }Z \\ \end{array}\]

Now we have to prove associativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Z,\text{ }Then, \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }a\text{ }*\text{ }\left( b\text{ }+\text{ }c\text{ }+\text{ }b\text{ }c \right) \\ =\text{ }a\text{ }+\text{ }\left( b\text{ }+\text{ }c\text{ }+\text{ }b\text{ }c \right)\text{ }+\text{ }a\text{ }\left( b\text{ }+\text{ }c\text{ }+\text{ }b\text{ }c \right) \\ =\text{ }a\text{ }+\text{ }b\text{ }+\text{ }c\text{ }+\text{ }b\text{ }c\text{ }+\text{ }a\text{ }b\text{ }+\text{ }a\text{ }c\text{ }+\text{ }a\text{ }b\text{ }c \\ \left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }=\text{ }\left( a\text{ }+\text{ }b\text{ }+\text{ }a\text{ }b \right)\text{ }*\text{ }c \\ =\text{ }a\text{ }+\text{ }b\text{ }+\text{ }a\text{ }b\text{ }+\text{ }c\text{ }+\text{ }\left( a\text{ }+\text{ }b\text{ }+\text{ }a\text{ }b \right)\text{ }c \\ =\text{ }a\text{ }+\text{ }b\text{ }+\text{ }a\text{ }b\text{ }+\text{ }c\text{ }+\text{ }a\text{ }c\text{ }+\text{ }b\text{ }c\text{ }+\text{ }a\text{ }b\text{ }c \\ =>a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }\left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c,\text{ }\forall \text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Z \\ \end{array}\]

Thus, * is associative on Z.

(ii) to check : commutativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }N \\ a\text{ }*\text{ }b\text{ }=\text{ }{{2}^{ab}} \\ =\text{ }{{2}^{ba}} \\ =\text{ }b\text{ }*\text{ }a \\ =>\text{ }a\text{ }*\text{ }b\text{ }=\text{ }b\text{ }*\text{ }a,\text{ }\forall \text{ }a,\text{ }b\text{ }\in \text{ }N \\ \end{array}\]

Thus, * is commutative on N

to check: associativity of *

Let a, b, c ∈ N

Then, a * (b * c) = a * (2bc)

=

2a∗2bc

(a * b) * c = (2ab) * c

=

2ab∗2c

=> a * (b * c) ≠ (a * b) * c

Thus, * is not associative on N

i

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