Login/Sign Up
Home » Check the commutativity and associativity of each of the following binary operations: (xiii) ‘*’ on Q defined by a * b = (ab/4) for all a, b ∈ Q (xiv) ‘*’ on Z defined by a * b = a + b – ab for all a, b ∈ Z
Check the commutativity and associativity of each of the following binary operations: (xiii) ‘*’ on Q defined by a * b = (ab/4) for all a, b ∈ Q (xiv) ‘*’ on Z defined by a * b = a + b – ab for all a, b ∈ Z
Binary Operations | CBSE | Class 12 | Exercise 3.2 | Maths | RD Sharma
Check the commutativity and associativity of each of the following binary operations: (xiii) ‘*’ on Q defined by a * b = (ab/4) for all a, b ∈ Q (xiv) ‘*’ on Z defined by a * b = a + b – ab for all a, b ∈ Z

(xiii) to check :commutativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }\left( ab/4 \right) \\ =\text{ }\left( ba/4 \right) \\ =\text{ }b\text{ }*\text{ }a \\ ~a\text{ }*\text{ }b\text{ }=\text{ }b\text{ }*\text{ }a,\text{ }for\text{ }all\text{ }a,\text{ }b\text{ }\in \text{ }Q \\ \end{array}\]

Thus, * is commutative on Q

to check: associativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Q,\text{ }then \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }a\text{ }*\text{ }\left( b\text{ }c/4 \right) \\ =\text{ }\left[ a\text{ }\left( b\text{ }c/4 \right) \right]/4 \\ =\text{ }\left( a\text{ }b\text{ }c/16 \right) \\ \left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }=\text{ }\left( ab/4 \right)\text{ }*\text{ }c \\ =\text{ }\left[ \left( ab/4 \right)\text{ }c \right]/4 \\ =\text{ }a\text{ }b\text{ }c/16 \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }\left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }for\text{ }all\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Q \\ \end{array}\]

Thus, * is associative on Q.

(xiv) to check: commutativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b\text{ }\in \text{ }Z,\text{ }then \\ a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }+\text{ }b\text{ }-\text{ }ab \\ =\text{ }b\text{ }+\text{ }a\text{ }-\text{ }ba \\ =\text{ }b\text{ }*\text{ }a \\ ~a\text{ }*\text{ }b\text{ }=\text{ }b\text{ }*\text{ }a,\text{ }for\text{ }all\text{ }a,\text{ }b\text{ }\in \text{ }Z \\ \end{array}\]

Thus, * is commutative on Z.

to check: associativity of *

    \[\begin{array}{*{35}{l}} Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Z \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }a\text{ }*\text{ }\left( b\text{ }+\text{ }c\text{ }-\text{ }b\text{ }c \right) \\ =\text{ }a\text{ }+\text{ }b\text{ }+\text{ }c-\text{ }b\text{ }c\text{ }\text{ }ab\text{ }-\text{ }ac\text{ }+\text{ }a\text{ }b\text{ }c \\ \left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }=\text{ }\left( a\text{ }+\text{ }b\text{ }-\text{ }a\text{ }b \right)\text{ }c \\ =\text{ }a\text{ }+\text{ }b\text{ }-\text{ }ab\text{ }+\text{ }c\text{ }\text{ }\left( a\text{ }+\text{ }b\text{ }-\text{ }ab \right)\text{ }c \\ =\text{ }a\text{ }+\text{ }b\text{ }+\text{ }c\text{ }-\text{ }ab\text{ }-\text{ }ac\text{ }-\text{ }bc\text{ }+\text{ }a\text{ }b\text{ }c \\ a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }\left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c,\text{ }for\text{ }all\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }Z \\ \end{array}\]

 

Thus, * is associative on Z.

i

More Material