Consider the binary operations * : R x R → R and o : R x R → R defined as a * b = |a - b| and a o b = a, ∀ a, b ∈ R. Show that * is commutative but not associative, o is associativ... Consider the binary operations * : R x R → R and o : R x R → R defined as a * b = |a - b| and a o b = a, ∀ a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a * (b o c) = (a * b) o (a * c). Does o distribute over *? Justify your answer. Read more

Steps to Solve

1. Show that * is Commutative

To show that the operation $*$ is commutative, we need to prove that for any $a, b \in \mathbb{R}$, $a * b = b * a$:

$$ a * b = |a - b| $$ $$ b * a = |b - a| = |a - b| \quad \text{(since absolute values are equal)} $$

Thus, $a * b = b * a$, confirming that $*$ is commutative.

2. Show that * is Not Associative

To show that $*$ is not associative, we need an example where $(a * b) * c \neq a * (b * c)$.

Let $a = 1$, $b = 2$, and $c = 3$:

Calculate $(a * b) * c$: $$ a * b = |1 - 2| = 1 $$ Then, $$ (a * b) * c = 1 * 3 = |1 - 3| = 2 $$

Now calculate $a * (b * c)$: $$ b * c = |2 - 3| = 1 $$ Then, $$ a * (b * c) = 1 * 1 = |1 - 1| = 0 $$

Since $2 \neq 0$, we conclude that $*$ is not associative.

3. Show that o is Associative

To prove that $o$ is associative, we need to show that for all $a, b, c \in \mathbb{R}$, $$ (a o b) o c = a o (b o c) $$

With the definition $a o b = a$:

Calculate $(a o b) o c$: $$ (a o b) o c = a o c = a $$

Calculate $a o (b o c)$: $$ b o c = b \quad \text{(since $o$ always returns the first argument)} $$ Thus, $$ a o (b o c) = a o b = a $$

Since both expressions equal $a$, the operation $o$ is associative.

4. Show that o is Not Commutative

To show that $o$ is not commutative, we need to find $a, b$ such that $a o b \neq b o a$.

Let $a = 1$ and $b = 2$: $$ 1 o 2 = 1 \quad \text{and} \quad 2 o 1 = 2 $$ Since $1 \neq 2$, $o$ is not commutative.

5. Show that a * (b o c) = (a * b) o (a * c)

Let $a, b, c \in \mathbb{R}$: Calculate $b o c$: $$ b o c = b $$ Hence, $$ a * (b o c) = a * b = |a - b| $$

Now calculate $(a * b) o (a * c)$: $$ a * b = |a - b| \quad \text{and} \quad a * c = |a - c| $$ Then, $$ (a * b) o (a * c) = |a - b| \quad \text{(as } o \text{ returns the first element)} $$

Thus, we have shown that: $$ a * (b o c) = (a * b) o (a * c) = |a - b| $$

6. *Does o Distribute Over ?

To check if $o$ distributes over $*$, we must establish: $$ a o (b * c) \stackrel{?}{=} (a o b) * (a o c) $$

Calculating each side:

• LHS: $$ a o (b * c) = a \quad \text{(since $o$ always returns the first argument)} $$
• RHS: $$ (a o b) * (a o c) = a * a = |a - a| = 0 $$

Since $a \neq 0$ generally, the two sides are not equal.