Question

Show that addition, subtraction, and multiplication are binary operations on $R$, but division is not a binary operation on $R$. Further, show that division is a binary operation on the set $R_{\ast }$ of nonzero real numbers.

Answer 1

Solve to prove that addition, subtraction, and multiplication are binary operations on $R$, but division is not.

For Addition,

$+:R\times R\to R$

where $(a,b)\to a+b$
For every real number $a\&b$,

$a+b$ is also a real number.

Hence, $+$ is a binary operation on $R$.

For Subtraction

$-:R\times R\to R$

where $(a,b)\to a-b$

For every real number $a\&b,a-b$ is also a real number.

Hence, $-$ is a binary operation on $R$.

For Multiplication,

$\times :R\times R\to R$

where $(a,b)\to a\times b$
For every real number $a\&b$,
$a\times b$ is also a real number.
Hence, $\times$ is a binary operation on $R$ .

For Division

$÷:R\times R\to R$
where $(a,b)\to a÷b$

Here, $a\&b$ are real numbers

$a÷b=\frac{a}{b}$

Let $a=3\&b=0$
$\frac{a}{b}=\frac{3}{0}$ "Not defined"
Hence, $÷$ is not a binary operation on $R$.

Solve to prove that division is a binary operation on the set $R_{\ast }$ of nonzero real numbers.

$÷:R_{\ast }\times R_{\ast }\to R_{\ast }$

" where $(a,b)\to a÷b$

For every non-zero real number $a\&b$

$a÷b$ is also a non-zero real number.

For example:$p=4$ and $q=-3$
$\frac{p}{q}=-\frac{4}{3}\in R_{\ast }$

Hence, $÷$ is a binary operation on $R_{\ast }$