Consider the binary operations ∗:R×R→R and o:R×R→R defined as a∗b=∣a−b∣ and aob=a for all a,b∈R. Show that ′∗′ is commutative but not associative, 'o' is associative but not commutative.

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Consider the binary operations : R R R and o : R R R define

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Consider the binary operations $* : R \times R \to R$ and $o : R \times R \to R$ defined as $a * b = ∣ a - b ∣$ and $a o b = a$ for all $a, b \in R$ . Show that $^{'} *^{'}$ is commutative but not associative, 'o' is associative but not commutative.

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Harper Discussed

Consider the binary operations

* : R \times R \to R

and

o : R \times R \to R

defined as

a * b = ∣ a - b ∣

and

a o b = a

for all

a, b \in R

. Show that

^{'} *^{'}

is commutative but not associative, 'o' is associative but not commutative.

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Two numbers are said to satisfy the commutative property if

p * q = q * p

They satisfy the associative property if

p * (q * r) = (p * q) * r

a * b = ∣ a - b ∣

b * a = ∣ b - a ∣ = ∣ a - b ∣

(a * b) * c = ∣ a - b ∣ * c = ∣∣ a - b ∣ - c ∣... (1)

a * (b * c) = a * ∣ b - c ∣ = ∣ a - ∣ b - c ∣∣... (2)

The expressions (1) and (2) might not be the same.

\Rightarrow^{'} *^{'}

is commutative but not associative.

a o b = a, b o a = b

(a o b) oc = a oc = a, a o (b oc) = a o b = a

\Rightarrow^{'} o^{'}

is associative but not commutative.

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Question Text	Consider the binary operations $* : R \times R \to R$ and $o : R \times R \to R$ defined as $a * b = ∣ a - b ∣$ and $a o b = a$ for all $a, b \in R$ . Show that $^{'} *^{'}$ is commutative but not associative, 'o' is associative but not commutative.
Answer Type	Text solution:1
Upvotes	151