Express set $[a,b] \cup [c,d]$ as the difference of two intervals

Question

So if $a < b < c < d$ , how do I express the set $[a,b] \cup [c,d]$ as the difference of two intervals? Is it as simple as $[a,d] -[b,c]$?

Answer 1

$$[a,d]\backslash(b,c)$$

$$\mathbb{R} = (-\infty, a) \cup [a,b] \cup (b,c) \cup [c,d] \cup (d, \infty)$$

If you imagine the union on a number line, it is $[a,d]$ without $(b,c)$

As $$[a,d] = [a,b] \cup (b,c) \cup [c,d] $$

And $$[a,d]\backslash(b,c) = ([a,b] \cup (b,c) \cup [c,d])\backslash(b,c)$$ Hence, $$[a,d]\backslash(b,c) = [a,b] \cup [c,d]$$

You mentioned that you thought the answer was: [a,d][b,c] Which is almost correct, however,

1) There is a difference between [b,c] and (b,c)

2) Mathematicians like to use '\' instead of '-' for set difference.

When '(' is used it means that you are not including the number next to it. i.e. (a,b] means a line from a to b, not containing a, but containing b.

$[a,d]\backslash[b,c]$ would mean $[a,b)\cup(c,d]$ - This is different from $[a,b]\cup[c,d]$