Are Integers Closed under Addition?
In mathematics, a set is said to be closed under an operation if the result of performing that operation on any two elements of the set is also an element of the set.
Integers
Integers are the set of whole numbers, both positive and negative, and zero. They are represented by the symbol ℤ.
Addition
Addition is a binary operation that takes two numbers and returns their sum.
Are Integers Closed under Addition?
Yes, integers are closed under addition.
To prove this, we need to show that the sum of any two integers is also an integer.
Let’s consider two integers, a and b.
If a and b are both positive, then their sum, a + b, is also positive.
If a and b are both negative, then their sum, a + b, is also negative.
If a and b have opposite signs, then their sum, a + b, is either positive or negative, depending on which number has the greater absolute value.
In all cases, the sum of two integers is an integer. Therefore, integers are closed under addition.
Conclusion
Integers are closed under addition because the sum of any two integers is also an integer.
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