FizeauV

2020-12-16

Let $×$ be a binary operation on set of rational number $\mathbb{Q}$ defined as follows: $a\cdot b=a+b+2ab$, where $a,b\in \mathbb{Q}$
a) Prove that $×$ is commutative, associate algebraic operation on $\mathbb{Q}$

Mitchel Aguirre

Expert

a) For commutative
Prove $a\cdot b=b\cdot a$
Now $a\cdot b=a+b+2ab$
and $b\cdot a=b+a+2ba$
$b\cdot a=a+b=a+b+2ab$

$⟨a+b=b+a\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}a\cdot b=b\cdot a⟩$
Hence $a\cdot b=b\cdot a$
For associatvie
Prove$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
Now$\left(a\cdot b\right)\cdot c=\left(a+b+2ab\right)\cdot c$
$=a+b+2ab+c+2\left(a+b+2ab\right)c$
$=a+b+2ab+2ac+abc+4abc$
and $a\cdot \left(b\cdot c\right)=a\cdot \left(b+c+2bc\right)$
$=a+b+c+2bc+2a\left(b+c+2bc\right)$
$=a+b+c+2bc+2ab+2ac+4abc$
$=a+b+c+2ab+2ac+2bc+4abc$
Hence $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

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