Exercise.
In a university 60% of students learn analysis, 50% learn algebra and 50% learn set theory.
We know that 30% learn analysis and algebra, 20% learn algebra and set theory, 40% learn analysis and set theory. 10% of students learn analysis, algebra and set theory.
What percentage of students
(a) do not learn any mentioned topic?
(b) study exactly two topics?
(c) study at least two topics?
Denote $A$ the set of students who learn analysis, $B$ the set of students who learn algebra, $C$ the set of students who learn set theory.
$\Longrightarrow$
$\Longrightarrow$
(a) $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$
$=80.$
(b) $|[(A\cap B)-C]\cup^*[(A\cap C)-B]\cup^*[(B\cap C)-A]|=60.$
(c) $|A\cap B\cap C|+\text{previous answer}=70.$
$\Longrightarrow$
$\Longrightarrow$
(a) $|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$
$=80.$
(b) $|[(A\cap B)-C]\cup^*[(A\cap C)-B]\cup^*[(B\cap C)-A]|=60.$
(c) $|A\cap B\cap C|+\text{previous answer}=70.$
