True or false the set of integers is closed under subtraction? Please give an explanation!!!
Please help ASAP!!! :(
if you subtract any integer from another, you get an integer.
So, what does that mean?
The set of integers is indeed closed under subtraction, which means that when you subtract two integers, the result will always be an integer. Here's an explanation:
To prove that the set of integers is closed under subtraction, we need to show that for any two integers, their difference is also an integer.
Let's consider two arbitrary integers, say a and b. Since a and b are integers, we can write them as a = m and b = n, where m and n are some whole numbers.
When we subtract b from a (a - b), we can substitute the values of a and b, yielding (m - n). Now, we need to determine if (m - n) is still a whole number (integer).
To do that, we need to consider a few cases:
1. If m > n: In this case, when we subtract two whole numbers, the result will always be a whole number. Therefore, (m - n) is an integer.
2. If m < n: In this situation, when we subtract two whole numbers, the result might not be a whole number. For example, if m = 2 and n = 5, then (m - n) = (2 - 5) = -3, which is still an integer.
3. If m = n: When the two integers are the same, their difference will be zero, which is also an integer.
In all three cases, we find that the difference between two integers, regardless of the specific values, is always an integer. Therefore, we can conclude that the set of integers is closed under subtraction.
Remember that this explanation proves that the set of integers is closed under subtraction in general, regardless of specific values chosen.