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Let S={a+sqrt(2)b : a , b in Z}dot Then...

Let `S={a+sqrt(2)b : a , b in Z}dot` Then prove that an operation * on S defined by `(a_1+sqrt(2)b_1)*(a_2+sqrt(2)b_2)=(a_1+b_2)+sqrt(2)(b_1+b_2)` for all `b_1,a_2 in Z` is binary operation ofn `Sdot`

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`(a_{1}+sqrt{2} b_{1}) *(a_{2}+sqrt{2} b_{2})`

` Rightarrow (a_{1}+a_{2}) in Z}+ sqrt{2}(b_{1}+b_{2}) in Z`

`((a_{1}+a_{2}) + sqrt{2}(b_{1}+b_{2}) )in S`
Hence,`*` is a binary opreation
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