Question

Which of the following are binary operations. Justify your answer.
i) The operation · defined on Q by a·b = a(b− 1).
ii) The operation · defined on [0,π] by x·y = cosxy.
Also, for those operations which are binary operations, check whether they are
associative and commutative.

Answer 1

Answer on Question #62050 – Math – Abstract Algebra

Question

Which of the following are binary operations. Justify your answer.

i) The operation $\cdot$ defined on $Q$ by $a \cdot b = a(b - 1)$.

ii) The operation $\cdot$ defined on $[0, \pi]$ by $x \cdot y = \cos xy$.

Also, for those operations which are binary operations, check whether they are associative and commutative.

Solution

A binary operation $f(x, y)$ is an operation that applies to two quantities or expressions $x$ and $y$.

A binary operation on a nonempty set $A$ is a map $f: A \times A \to A$ such that

1. $f$ is defined for every pair of elements in $A$, and

2. $f$ uniquely associates each pair of elements in $A$ to some element of $A$.

i) The operation $\cdot$ defined on $Q$ by $a \cdot b = a(b - 1)$ is binary

The operation is defined for all $a, b \in Q$. For any $a, b \in Q$, $a(b - 1) \in Q$.

If $a = a_1$ and $b = b_1$, hence $a(b - 1) = a_1(b_1 - 1)$.

There exist $a$ and $b$ in $Q$ such that $a(b - 1) \neq (a - 1)b$, hence the operation is not commutative. For example: $4(\frac{3}{5} - 1) \neq \frac{3}{5}(4 - 1)$.

There exist $a$, $b$ and $c$ in $Q$ such that $a(b(c - 1) - 1) \neq (a(b - 1))(c - 1)$, hence the operation is not associative. For example: $\frac{3}{5}(2(3 - 1) - 1) \neq \frac{3}{5}(2 - 1)(3 - 1)$.

Answer: The operation $\cdot$ defined on $Q$ by $a \cdot b = a(b - 1)$ is binary, not associative and not commutative.

ii) The operation $\cdot$ defined on $[0, \pi]$ by $x \cdot y = \cos xy$ is not binary. Because there exist $a$ and $b$ in $[0, \pi]$ such that $\cos ab \notin [0, \pi]$.

For example, $\cos \pi \cdot 1 = -1 \notin [0, \pi]$.

Answer: The operation $\cdot$ defined on $[0, \pi]$ by $x \cdot y = \cos xy$ is not binary.

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