Topic
Find the sets \(A = \left[ { – 1;7} \right],B = \left( {m – 1;m + 5} \right)\) where m is a real parameter. Find m to
a) \(B \subset A\)
b) \(A \cap B = \emptyset \)
Detailed explanation
a)
To \(B \subset A\) then \(\left\{ {\begin{array}{*{20}{c}}{m – 1 \ge – 1}\\{m + 5 \le 7} \end{array} \Leftrightarrow \left\{ {\begin{array}{*{20}{c}}{m \ge 0}\\{m \le 2}\end{array}} \right.} \right.\Leftrightarrow 0 \le m \le 2\)
So for m satisfying \(0 \le m \le 2\) then \(B \subset A\)
b)
To \(A \cap B = \emptyset \) then \(\left[{\begin{array}{*{20}{c}}{m–1\ge7}\\{m+5\le–)1}\end{array}}\right\Leftrightarrow\left[{\begin{array}{*{20}{c}}{m\ge8}\\{m\le–6}\end{array}}\right\)[{\begin{array}{*{20}{c}}{m–1\ge7}\\{m+5\le –1}\end{array}}\right\Leftrightarrow\left[{\begin{array}{*{20}{c}}{m\ge8}\\{m\le –6}\end{array}}\right\)
So for m satisfying \(m \le – 6\) or \(m \ge 8\) then \(A \cap B = \emptyset \)
