Question

Assertion (A) The binary operation $*: R \times R \rightarrow R$ given by $a * b \rightarrow a+2 b$ is associative.
Reason (R) A binary operation $*: A \times A \rightarrow A$ is said to be associative, if $(a * b) * c=a *(b * c)$ 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
$(8 * 5) * 3=(8+10) * 3=(8+10)+6=24,$
while $8 *(5 * 3)=8 *(5+6)=8 * 11=8+22=30$
Note Associative property of a binary operation is very important in the sense that with this property of a binary operation, we can write $a_1 * a_2 * \ldots * a_n$ which is not ambiguous. But in absence of this property, the expression $a_1 * a_2 * \ldots * a_n$ 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.