The union of two sets produces a new set which consists of all the elements from both sets “dumped together” into one big set (without duplicates).
A\cup B=\{1,2,3,5,7, 0,2,4,6,8\}
Arrangement of elements in the set doesn’t matter. All we are doing is taking all the elements of A and putting them together with the elements of B in one big set.
A\cup B=\{-3,-4,-5,0,1, 2, 5\}
Notice that A and B have elements in common. So, when A and B are “dumped together” into one big set, you don’t need to list the duplicate elements twice, because sets don’t contain duplicates.
A\cup B=\{0,1,2,3,4,5\}
Notice that A\subseteq B, so dumping the sets together and eliminating duplicate elements gives you exactly the set B.
A\cup \emptyset=\{1,2,3\}=A.
Think of it this way: you have a box containing 1,2 and 3, and you dump that together with a box that contains nothing. Would you get anything more than what was in the first set?
\emptyset\cup\emptyset=\emptyset.
Think of it this way: if you dump two boxes that contain nothing together into one big box, what do you end up with (in that big box)?
(0,3)\cup(2,5)=(0,5).
Solution: (0,3)\cup(2,5)=(0,5). The given two intervals (0,3) and (2,5) overlap on the interval (2,3), so the list of elements that are in (0,3) OR (2,5) are all elements between 0 and 3, non-inclusive of endpoints. See number line below. Everything shaded is in the set (0,3)\cup(2,5).
(-1,2)\cup (3,6) cannot be simplified any more than it is.
The two intervals listed do not overlap at all, so all numbers between 2 and 3 (including 2 and 3 themselves) are NOT in the set (-1,2)\cup (3,6). This set is drawn below on the number line. Everything that is shaded is in the set.
(-1,4)\cup [4,6]=(-1,6].
While the intervals (-1,4) and [4,6] do not overlap at all, the interval (-1,4) ends where the interval [4,6] starts. In other words, (-1,4) contains all numbers greater than -1 up to 4, and [4,6] contains all numbers from 4 to 6, INCLUDING 4. Because of this “one set picking up where the other left off”, we can write this union as a single interval. NOTE: if 4 was NOT included in the second interval, we could not have written this union as a single interval; there would have been a “gap” between the two sets occurring at 4. See number line below. Everything that is shaded lives in the union.
Can’t be simplified any more, so (-\infty, 0)\cup (0,\infty)
Solution: Can’t be simplified any more. While the first interval stops where the second one starts, neither set contains the element 0, so one cannot write (-\infty, 0)\cup (0,\infty)=(-\infty,\infty), because the set on the right-hand side of the equation contains 0, whereas the one on the left-hand side does not!