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Let \(\ast\) be a binary operation on the set \(Q\) of rational numbers as follows: $\;\; a \ast b = a-b$. Find which of the binary operations are commutative and which are associative.

Note: This is part 1 of a 6 part question, split as 6 separate questions here.

1 Answer

  • An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$
  • An operation $\ast$ on $A$ is associative if $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
Given a binary operation $\ast$ in Q defined by $a \ast b=a-b$:
$\textbf {Step 1: Checking if the operation is Commutative}$:
For an operation $\ast$ to be commutative $a\ast b = b \ast a$.
Consider $a=1, b=2 \rightarrow a \ast b = 1 - 2 = -1$, and $b \ast a = 2 - 1 = 1$.
Since $a\ast b \neq b \ast a$, $\ast$ is not commutative.
$\textbf {Step 2: Checking if the operation is Associative}$:
For an operation $\ast$ on $A$ is associative $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
Consider $a=1,b=2, c=3 \rightarrow a \ast (b \ast c) = a \ast (2-3) = 1 \ast (-1) = 2$
$(a \ast b) \ast c = (1-2) \ast 3 = 1-2-3 = -4$.
Since $a\ast ( b \ast c) \neq (a \ast b) \ast c \; \ast$ is not associative
answered Mar 19, 2013 by balaji.thirumalai

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