Question

# Let * be a binary operation on set of rational number QQ defined as follows:a*b=a+b+2ab, where a,b in QQa) Prove that * is commutative, associate algebraic operation on QQ

Commutative Algebra

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

2020-12-17

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

$$\displaystyle{\left\langle{a}+{b}={b}+{a}{\quad\text{and}\quad}{a}\cdot{b}={b}\cdot{a}\right\rangle}$$
Hence $$a\cdot b=b\cdot a$$
For associatvie
Prove$$(a\cdot b)\cdot c=a\cdot(b\cdot c)$$
Now$$(a\cdot b)\cdot c=(a+b+2ab)\cdot c$$
$$=a+b+2ab+c+2(a+b+2ab)c$$
$$=a+b+2ab+2ac+abc+4abc$$
and $$a\cdot (b\cdot c)=a\cdot (b+c+2bc)$$
$$=a+b+c+2bc+2a(b+c+2bc)$$
$$=a+b+c+2bc+2ab+2ac+4abc$$
$$=a+b+c+2ab+2ac+2bc+4abc$$
Hence $$(a\cdot b)\cdot c=a\cdot(b\cdot c)$$