Question

Assertion (A) The binary operation $\ast :R\times R\to R$ given by $a\ast b\to a+2b$ is associative.
Reason (R) A binary operation $\ast :A\times A\to A$ is said to be associative, if $(a\ast b)\ast c=a\ast (b\ast c)$ for all $a,b,c\in A$.

• A. Both A and R are correct, R is the correct explanation of A
• B. Both $A$ and $R$ are correct, $R$ is not the correct explanation of $A$
• C. A is correct, $R$ is incorrect
• D. $R$ is correct, $A$ is incorrect

Answer 1

Answer: (D)

The operation * is not associative, since
$(8\ast 5)\ast 3=(8+10)\ast 3=(8+10)+6=24,$
while $8\ast (5\ast 3)=8\ast (5+6)=8\ast 11=8+22=30$
Note Associative property of a binary operation is very important in the sense that with this property of a binary operation, we can write $a_1\ast a_2\ast \dots \ast a_n$ which is not ambiguous. But in absence of this property, the expression $a_1\ast a_2\ast \dots \ast a_n$ is ambiguous unless brackets are used. Recall that in the earlier classes brackets were used whenever subtraction or division operations or more than one operation occurred.