Explain whether the following given are closed under addition:
{x|x �¸ W and x>100}
To determine whether the given set is closed under addition, we need to check if adding any two elements from the set will always result in a new element that also belongs to the set.
Let's consider the set {x | x is a real number and x > 100}.
Suppose we take two elements from the set, let's say x1 and x2, such that x1 > 100 and x2 > 100.
To check if the given set is closed under addition, we need to verify if x1 + x2 is also greater than 100.
If we add x1 + x2 and the result is greater than 100, then the set is closed under addition for these two particular elements.
For example, let x1 = 150 and x2 = 120. If we add these two numbers, we get 270, which is greater than 100. Thus, in this case, the set is closed under addition for x1 and x2.
However, if we choose x1 = 90 and x2 = 120, the sum would be 210, which is not greater than 100. Therefore, the set is not closed under addition for these particular elements.
Since there are cases where the set is closed under addition (e.g., x1 = 150 and x2 = 120), and cases where it is not (e.g., x1 = 90 and x2 = 120), we can conclude that the given set {x | x is a real number and x > 100} is not closed under addition.