Please help me solve these questions. Thank you very much. For this Big Problem, we’ll investigate Inclusion-Exclusion and its variations, and do some practice with set techniques along the way. We covered Inclusion-Exclusion a little bit in the textbook already. It might help to go back and read Section 2.3 and take a look at the exercises, too. (1) There are a couple of different ways to see why it’s true that |A| + |B| − |A ∩ B| = |A ∪ B|. Let’s start with one using membership tables: Write out the membership table for A, B, A ∩ B, and A ∪ B. (2) For each row in your membership table, compare the sum of the entries in A and B minus the entry in A ∩ B with the entries in A ∪ B. What do you notice? (3) What does this have to do with Inclusion-Exclusion? (4) Now let’s try thinking about Inclusion-Exclusion in terms of Venn diagrams. Draw a Venn diagram of A and B, and color in A and B in two different colors / shadings. (5) What do you notice about how A ∩ B is colored/shaded, compared to the rest of A ∪ B? (6) What does this have to do with Inclusion-Exclusion? (7) Give another reason why Inclusion-Exclusion makes sense. Your answer can be in any style including a specific example, a written explanation, or a diagram.
Please help me solve these questions. Thank you very much.
For this Big Problem, we’ll investigate Inclusion-Exclusion and its variations, and do some
practice with set techniques along the way. We covered Inclusion-Exclusion a little bit in
the textbook already. It might help to go back and read Section 2.3 and take a look at the
exercises, too.
(1) There are a couple of different ways to see why it’s true that |A| + |B| − |A ∩ B| =
|A ∪ B|. Let’s start with one using membership tables: Write out the membership
table for A, B, A ∩ B, and A ∪ B.
(2) For each row in your membership table, compare the sum of the entries in A and B
minus the entry in A ∩ B with the entries in A ∪ B. What do you notice?
(3) What does this have to do with Inclusion-Exclusion?
(4) Now let’s try thinking about Inclusion-Exclusion in terms of Venn diagrams. Draw
a Venn diagram of A and B, and color in A and B in two different colors / shadings.
(5) What do you notice about how A ∩ B is colored/shaded, compared to the rest of
A ∪ B?
(6) What does this have to do with Inclusion-Exclusion?
(7) Give another reason why Inclusion-Exclusion makes sense. Your answer can be in
any style including a specific example, a written explanation, or a diagram.
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