Question

Consider the binary operations`*: RxxR->R` and `o: RxxR->R` defined as `a*b=|a-b|` and `aob=a` for all `a , b in Rdot` Show that `*` is commutative but not associative, `o` is associative but not commutative. Further, show that `*` is distributive over `o` . Dose `o` distribute over `*` ? Justify your answer.

Answer 1

In binary operation * `: R xx R rarr a ** b =|a -b|`
Let a,` b in R`
`therefore " " a ** (b ** c )=a ** |b-c|`
`= |a-|b-c|`
and (a * b) * c = | a - b | * c
= || a - b | - c|
`therefore` a * (b * c) c1 `ne` :- (a * b) * c (if a=1, b=2 ,c=3)
`rArr **` is not associative
In binary operation 0: `Rxx R rarr R, ` aob =a
Let `a,b in R`
`therefore aob =a`
and boa =b
`therefore aob ne boa `
`rArr ` o is not commuatative . Again let a, b,c `in R`
`therefore ao=(boc) = aob ne boa `
=a
and (aob)oc =aoc =a
`therefore` ao (boc) = aob
`rArr` o is associative
Now ,let a,b,c `in` R then
`therefore` (a * b) o (a * c) = (|a - b |) o (|a - c |) = | a - b |
`therefore` a * (boc) = a * b = | a - b |
`therefore` a * (boc) = (a * b) o (a* c)
`:.` Operation*, distributes on o.
Again 1 o (2 * 3) = 1 o ( |2 - 3 |) = 1 o 1 = 1
and (102) * (103) = 1 * 1 = | 1 - 1 | = 0
1 o (2 * 3 ) cp (lo2) * (103)
Operation o does not distribute on *.