Multiplication/addition of Intervals

Question

Let

$M=[0,2) \cup \{3\}$, $N=[4,9]$

Determine 3M, M+M, 2N, M+N

So I have no clue how to determine these, would $2N$ be $[8,18]$? Could someone help me? I can not find similar examples.

Answer 1

Yeah, it's as easy as you think it is.

$2N = \{2n| n \in N\} = \{k=2n| n\in [4,9]\}=$

$\{k=2n| 4 \le n \le 9\} = \{k= 2n| 8 \le 2n \le 18\}$

$= \{k| 8 \le k\le 18\} = [ 8, 18]$.

...

$M+N$ con be tricky though.

$M + N = \{m + n|m \in M; n \in N\} = $

$\{m+n| 0\le m \le 2$ or $m=3$ and $4\le n\le 9\}=$

$\{k| k = m+n; 0\le m \le 2; 4\le n \le 9\} \cup \{k|k = m+n; m = 3; 4\le n\le 9\}=$

$\{k| k=m+n; 4 \le m+n \le 11\}\cup \{k|k = n+3; 4\le n \le 9\}=$

$\{k| 4\le k \le 11\} \cup \{k| 7 \le k \le 12\} =$

$[4,11]\cup [7\cup 12] = [4, 12]$.