Question

Show that subtraction and division are not binary operations on $N$.

Answer 1

Solve to prove for subtraction.

$-:N\times N\to N$

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

We know that the set of natural numbers

$N=\{1,2,3,...\}$

Let $a\&b$ are natural numbers

Let $a=1\&b=4$

$a-b=1-4=-3\in N$

Hence, subtraction is not a binary operation on $N$.

Solve to prove for division.

$÷:N\times N\to N$

where $(a,b)\to a÷b$

We know that the set of natural numbers

$N=\{1,2,3,...\}$

Let $a\&b$ are natural numbers

$a÷b=\frac{a}{b}$

Let $a=2\&b=7$

$\frac{a}{b}=\frac{2}{7}\in N$

Hence, subtraction is not a binary operation on $N$.