Closure Under Operations

Today we will understand what is meant by the term Closure under a particular operation.

Let us consider the operation, $multiplication$ and operands of type $integers$. What do we get when we multiply any two integers? The result is always another integer. We cannot get a fraction by multiplying two integers.

Therefore, we say $\unicode{x201C}Integers\ are\ \underline{closed}\ under\ multiplication\unicode{x201D}$.

Now let us consider the operation $division$ and operands of type $integers$.

What happens when we divide one integer by another integer. We may get an integer or we may get a fraction, for example,

$5 \div 2 = 2\frac{1}{2} = 2.5$, which is a fraction.

Therefore, we can say, $\unicode{x201C}Integers\ are\ \underline{not\ closed}\ under\ division\unicode{x201D}$

We will see few more examples of closure.

$Natural\ numbers\ are\ closed\ under\ addition\ but\ not\ under\ subtraction.$

We know that natural numbers consist of all integers ranging from $1$ to $infinity$.

If we take any two natural numbers and add them we will always get another integer which is greater than $0$. Therefore natural numbers are closed under addition. On the contrary if we take two natural number and subtract one from the other, we could end up with a $0$ or a $negative$ number, for example:

$11 - 11 = 0$, or

$13 - 19 = -6$

Neither $0$ or $-6$ are natural numbers.

Therefore, we can say that natural numbers are not closed under subtraction.

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$Natural\ numbers\ are\ closed\ under\ multiplication\ but\ not\ under\ division.$

Similar to the above addition and subtraction concept, when we multiply two natural numbers, we always get another natural number as a result. However, when we divide a natural number by another natural number we may get a fraction as a result.

Therefore, natural numbers are closed under multiplication but not under division.

Remember, for some type of numbers (operands) to be closed under a particular operation, $ALL$ possible results must be of the same type.

For example, in case of whole numbers $20 \div 4 = 5$, here the operands, and the result are whole number, but we still cannot say that whole numbers are closed under division, since there are cases like $15 \div 2 = 7\dfrac{1}{2}$ where the result is not a whole number.