Interval Notation: (A U B) - C = (4,9)

[Steverino777]

Homework Statement

Let U= all real numbers, A=[2,9), B=(0,1], C=[-1,4]. Express in interval notation: (A union B) - C

Homework Equations

A union B includes all elements that are in either A or B, including any objects that happen to lie in both A and B.

The difference A - B consists of all objects that are elements of A and are not elemnts of B

The Attempt at a Solution

I think the interval is from (4,9). A union B would be (0,1]union[2,9], then you remove from [-1, 4]. The only reason why I don't know if I'm right is because of the gap in the union of A and B. I don't know if that has any effect on the answer.

If (4,9) is correct then would it be safe to assume that (A union B) - C = A-C ?

 

[CompuChip]

Your reasoning is correct.
Indeed in this case, $$(A \cup B) - C = A - C$$, because B is completely contained within C. So whatever elements from B you add to A when taking the union, you take them out again when removing C.
Of course, this is in general not true (e.g. A = [0, 1], B = [1, 2], C = [0, 1/2) is a counter example).

The gap in the union doesn't matter: X - Y is defined as the set of all elements which are in X but not in Y.