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Show that subtraction is not a binary operation on all natural numbers $N$.

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: In order to this question, to show whether the subtraction are not a binary operation on natural number $N$ , we will show the given statement by taking two natural number in which first number is smaller than the second number and when it will give the negative result, then it proves that subtraction are not binary operation on natural number $N$.

Complete step-by-step solution:
As we know that, the multiplication of natural numbers is also a natural number:
\[N \times N = N\]
Where, \[\left( {a,b} \right) = a - b\]
here a and b are natural numbers \[\left( {i.e.1,2,3,4,5,6,7,8,9...............} \right)\]
Now,
let \[a = 3\] and \[b = 5\]
\[a - b = 3 - 5\]
but since \[ - 2\] is not a natural number
\[\therefore \] Subtraction is not a binary operation on $N$ (natural number).

Note: The term "binary" refers to something that is made up of two parts. A binary operation is nothing more than a law for combining two values to produce a new one. The most well-known binary operations are addition, subtraction, multiplication, and division on different sets of numbers, which were taught in elementary school.
A binary operation on a set is a calculation that involves two set elements to generate another set element.
Natural numbers, which include all positive integers from 1 to infinity, are a part of the number system. Natural numbers, which do not contain zero or negative numbers, are also known as counting numbers. They are just positive integers, not zero, fractions, decimals, or negative ones, and they are a part of real numbers.