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Check the commutativity and associativity of each of the following binary operations: (vii) ‘*’ on Q defined by a * b = a + a b for all a, b ∈ Q (viii) ‘*’ on R defined by a * b = a + b -7 for all a, b ∈ R
Check the commutativity and associativity of each of the following binary operations: (vii) ‘*’ on Q defined by a * b = a + a b for all a, b ∈ Q (viii) ‘*’ on R defined by a * b = a + b -7 for all a, b ∈ R
Binary Operations | CBSE | Class 12 | Exercise 3.2 | Maths | RD Sharma
Check the commutativity and associativity of each of the following binary operations: (vii) ‘*’ on Q defined by a * b = a + a b for all a, b ∈ Q (viii) ‘*’ on R defined by a * b = a + b -7 for all a, b ∈ R

(vii)  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\text{ }+\text{ }ab  \\

b\text{ }*\text{ }a\text{ }=\text{ }b\text{ }+\text{ }ba  \\

=\text{ }b\text{ }+\text{ }ab  \\

~a\text{ }*\text{ }b\text{ }\ne \text{ }b\text{ }*\text{ }a  \\

\end{array}\]

Thus, * is not commutative on Q.

to prove : associativity on Q.

\[\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{ }+\text{ }b\text{ }c \right)  \\

=\text{ }a\text{ }+\text{ }a\text{ }\left( b\text{ }+\text{ }b\text{ }c \right)  \\

=\text{ }a\text{ }+\text{ }ab\text{ }+\text{ }a\text{ }b\text{ }c  \\

\left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }=\text{ }\left( a\text{ }+\text{ }a\text{ }b \right)\text{ }*\text{ }c  \\

=\text{ }\left( a\text{ }+\text{ }a\text{ }b \right)\text{ }+\text{ }\left( a\text{ }+\text{ }a\text{ }b \right)\text{ }c  \\

=\text{ }a\text{ }+\text{ }a\text{ }b\text{ }+\text{ }a\text{ }c\text{ }+\text{ }a\text{ }b\text{ }c  \\

~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.

(viii)  to check: commutativity of *

\[\begin{array}{*{35}{l}}

Let\text{ }a,\text{ }b\text{ }\in \text{ }R,\text{ }then  \\

a\text{ }*\text{ }b\text{ }=\text{ }a\text{ }+\text{ }b\text{ }-\text{ }7  \\

=\text{ }b\text{ }+\text{ }a\text{ }-\text{ }7  \\

=\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{ }R  \\

\end{array}\]

Thus, * is commutative on R

to prove : associativity of * on R.

\[\begin{array}{*{35}{l}}

Let\text{ }a,\text{ }b,\text{ }c\text{ }\in \text{ }R,\text{ }then  \\

a\text{ }*\text{ }\left( b\text{ }*\text{ }c \right)\text{ }=\text{ }a\text{ }*\text{ }\left( b\text{ }+\text{ }c\text{ }\text{ }7 \right)  \\

=\text{ }a\text{ }+\text{ }b\text{ }+\text{ }c\text{ }-7\text{ }-7  \\

=\text{ }a\text{ }+\text{ }b\text{ }+\text{ }c\text{ }\text{ }-14  \\

\left( a\text{ }*\text{ }b \right)\text{ }*\text{ }c\text{ }=\text{ }\left( a\text{ }+\text{ }b\text{ }\text{ }7 \right)\text{ }*\text{ }c  \\

=\text{ }a\text{ }+\text{ }b\text{ }\text{ }-7\text{ }+\text{ }c\text{ }\text{ }-7  \\

=\text{ }a\text{ }+\text{ }b\text{ }+\text{ }c\text{ }\text{ }-14  \\

a\text{ }*\text{ }\left( b\text{ }*\text{ }c\text{ } \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{ }R  \\

\end{array}\]

Thus, * is associative on R.

 

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