Login/Sign Up
Home » Check the commutativity and associativity of each of the following binary operations: (v) ‘o’ on Q defined by a o b = (ab/2) for all a, b ∈ Q (vi) ‘*’ on Q defined by a * b = ab2 for all a, b ∈ Q
Check the commutativity and associativity of each of the following binary operations: (v) ‘o’ on Q defined by a o b = (ab/2) for all a, b ∈ Q (vi) ‘*’ on Q defined by a * b = ab2 for all a, b ∈ Q
Binary Operations | CBSE | Class 12 | Exercise 3.2 | Maths | RD Sharma
Check the commutativity and associativity of each of the following binary operations: (v) ‘o’ on Q defined by a o b = (ab/2) for all a, b ∈ Q (vi) ‘*’ on Q defined by a * b = ab2 for all a, b ∈ Q

(v) to check: commutativity of o

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }o\text{ }b\text{ }=\text{ }\left( ab/2 \right) \\ =\text{ }\left( b\text{ }a/2 \right) \\ =\text{ }b\text{ }o\text{ }a \\ =>\text{ }a\text{ }o\text{ }b\text{ }=\text{ }b\text{ }o\text{ }a,\text{ }\forall \text{ }a,\text{ }b\text{ }\in \text{ }Q \\ \end{array}\]

Thus, o is commutative on Q

to check: associativity of o

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }o\text{ }\left( b\text{ }o\text{ }c \right)\text{ }=\text{ }a\text{ }o\text{ }\left( b\text{ }c/2 \right) \\ =\text{ }\left[ a\text{ }\left( b\text{ }c/2 \right) \right]/2 \\ =\text{ }\left[ a\text{ }\left( b\text{ }c/2 \right) \right]/2 \\ =\text{ }\left( a\text{ }b\text{ }c \right)/4 \\ \left( a\text{ }o\text{ }b \right)\text{ }o\text{ }c\text{ }=\text{ }\left( ab/2 \right)\text{ }o\text{ }c \\ =\text{ }\left[ \left( ab/2 \right)\text{ }c \right]\text{ }/2 \\ =\text{ }\left( a\text{ }b\text{ }c \right)/4 \\ =>\text{ }a\text{ }o\text{ }\left( b\text{ }o\text{ }c \right)\text{ }=\text{ }\left( a\text{ }o\text{ }b \right)\text{ }o\text{ }c,\text{ }\forall \text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Q \\ \end{array}\]

Thus, o is associative on Q.

(vi) to check : commutativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }a{{b}^{2}} \\ b\text{ }*\text{ }a\text{ }=\text{ }b{{a}^{2}} \\ a\text{ }*\text{ }b\text{ }\ne \text{ }b\text{ }*\text{ }a \\ \end{array}\]

Thus, * is not commutative on Q

Now we have to check associativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }a\text{ }*\text{ }\left( b{{c}^{2}} \right) \\ =\text{ }a\text{ }{{\left( b{{c}^{2}} \right)}^{2}} \\ =\text{ }a{{b}^{2}}~{{c}^{4}} \\ \left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }=\text{ }\left( a{{b}^{2}} \right)\text{ }*\text{ }c \\ =\text{ }a{{b}^{2}}{{c}^{2}} \\ ~a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }\ne \text{ }\left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c \\ \end{array}\]

Thus, * is not associative on Q.

i

More Material