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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| − |An B| = AU B. Let's start with one using membership tables: Write out the membership table for A, B, An B, and AUB. (2) For each row in your membership table, compare the sum of the entries in A and B minus the entry in AB with the entries in AUB. 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 An B is colored/shaded, compared to the rest of AUB? (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.

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| − |An B| = AU B. Let's start with one using membership tables: Write out the membership table for A, B, An B, and AUB. (2) For each row in your membership table, compare the sum of the entries in A and B minus the entry in AB with the entries in AUB. 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 An B is colored/shaded, compared to the rest of AUB? (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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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| – |An B| = AU B. Let's start with one using membership tables: Write out the membership table for A, B, An B, and AUB. (2) For each row in your membership table, compare the sum of the entries in A and B minus the entry in AB with the entries in AUB. 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 An B is colored/shaded, compared to the rest of AUB? (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.
Transcribed Image Text: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| – |An B| = AU B. Let's start with one using membership tables: Write out the membership table for A, B, An B, and AUB. (2) For each row in your membership table, compare the sum of the entries in A and B minus the entry in AB with the entries in AUB. 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 An B is colored/shaded, compared to the rest of AUB? (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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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| – |An B| = AU B. Let's start with one using membership tables: Write out the membership table for A, B, An B, and AUB. (2) For each row in your membership table, compare the sum of the entries in A and B minus the entry in AB with the entries in AUB. 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 An B is colored/shaded, compared to the rest of AUB? (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.
Transcribed Image Text: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| – |An B| = AU B. Let's start with one using membership tables: Write out the membership table for A, B, An B, and AUB. (2) For each row in your membership table, compare the sum of the entries in A and B minus the entry in AB with the entries in AUB. 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 An B is colored/shaded, compared to the rest of AUB? (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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