whole numbers are closed under the operation of addition, explain the statement with example
Answers
Two whole numbers add up to give another whole number. This is the closure property of the whole numbers. It means that the whole numbers are closed under addition. If a and b are two whole numbers and a + b = c, then c is also a whole number. 3 + 4 = 7 (whole number).
Step-by-step explanation:
We know from arithmetic that a set is closed under an operation if the output of that operation on members of the set always produces a member of that set. Let us define a binary operation o which operates between the elements of the set A, then we say o satisfies the closure property or set A is closed under the operation o if aob∈A for alla,b∈A.
The whole number set W takes all the natural numbers and the number zero. So we have
W={0,1,2,3,}
If we take any two elements from the whole number set and add them we will get the sum also a whole number, for example, we take 0 and 1 then the sum will be0+1=1. So we can say for all a,b∈W we will find a+b∈W. It means the whole number set is closed under the operation addition. So option A is not correct.
If we take any two elements from the whole number set and subtract one from the other we may not get a whole number, for example, 0−1=−1 where the result −1 is outside the whole number set in the set of integers. We can never get a whole number when we subtract greater numbers from the smaller number. It means if b>a, a−b∉W. So the whole number set is not closed under subtraction and option B is correct.
If we take any two elements from the whole number set and multiply them we will get the product also a whole number, for example, 0×1=0 where 0 is the whole number. So we can say for all a,b∈W we will find a×b∈W. It means the whole number set is closed under the operation multiplication.