Question

Simple? Set Theory Question.
For the collection given below, determine whether the set of the collection is mutually disjoint. Also, determine whether the collection is nested, and find the union and intersection.

Answer 1

\[D={(x,)|x \in \mathbb{R}}\]

Answer 2

suppose to say (x, infinity)

Answer 3

So this is giving you the set of all intervals \((x, \infty)\) such that \(x\in\mathbb{R}\) correct?

Answer 4

yes. that is correct.

Answer 5

so i am thinking to start with some really big number, say -100,000, go up like 10 or so, try the other direction and look for patterns, but not sure if that will be conclusive enough.

Answer 6

Just look at these two elements in the set. \[(0,\infty),\;(1,\infty)\]Are these mutually disjoint?

Answer 7

no, they are not. for 1,infinity is inside of 0,infinity.

Answer 8

so it is also nested though, since the latter is in the former? and the same would go for all above it or below it?

Answer 9

union would be (-inf,inf) and intersection still is a bit confusing...

Answer 10

All right so far.

Answer 11

So the intersection is merely the set of number in every single one of these sets. So suppose \[a\in\bigcap_{x\in\mathbb{R}}(x,\infty)\]However, \(a\notin(a+1,\infty)\). What does this tell you about the intersection?

Answer 12

it would be the empty set then, right? that was my suspicion, but was unsure how to show it. :)

Answer 13

That's precisely it. The intersection would be empty, for if it was not, we would get a contradiction.

Answer 14

haha. thanks for the help. :)

Answer 15

You're welcome.