Question

Consider the binary operations

$\ast :R\times R\to Rando:R\times R\to R$ defined as $a\ast b=|a-b|$ and $a\circ b=a,\forall a,b\in R.$Show that * is commutative but not associative, o is associative but not commutative. Further, show that $\forall a,b,c\in R,a\ast (b\circ c)=(a\ast b)\circ (a\ast c)$. (If it is so, we say that the operation * distributes over the operation o). Does o distribute over? Justify your answer.

Answer 1

Checking commutative for *
Given : $a\ast b=|a-b|\&aob=a,\forall a,b\in R$
* is commutative if,
a * b = b * a
Now,
a * b = |a - b|
And,
b * a = |b - a|
= |a - b|
Since,

$a\ast b=b\ast a\forall a,b\in R$
* is commutative.
Checking associative for *
* is associative if,
(a * b) * c = a * (b * c)
Now,
(a * b) * c = (|a - b|) * c
= ||a - b|- c|
And,
a * (b * c) = a * (|b - c|)
= |a - |b - c||
Since,
$(a\ast b)\ast c\ne a\ast (b\ast c)$
* is not associative.
Check commutative for o.
Given: a o b = a
o is commutative if,
a o b = a and b o a = b
Since $aob\ne boa,$
o is not commutative.

Checking associative for o
o is associative if,
(a o b) o c = a o (b o c)
Now,
(a o b)o c = a o c
= a
And,
a o (b o c) = a o b
= a
Since (a o b) o c = a o (b o c)
o is an associative.
* distributes over o
* distrubutes over o if a * (b o c) =
(a * b) o (a * c), $\forall a,b,c\in R$
Now,
a * (boc) = a * b
= |a - b|
And,
(a * b) o (a * c) = |a - b|o|a - c| = |a - b|
Since
a * (boc) = (a* b) o (a * c), $\forall a,b,c\in R$
$\therefore$ * distributes over o.

o distributes over *
o distributes over * If,
ao ( b * c) = (aob) * (aoc), $\forall a,b,c\in R$
Now,
a o (b * c) = a o |b - c| = a
(a o b) * (a o c) = a * a
= |a - a|
= |0| = 0
Since,
$a\circ (b\ast c)\ne (a\circ b)\ast (a\circ c)$
o does not distribute over *