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For this Big Problem, we're going to investigate induction in different settings, and think about when it does and doesn't work. (1) The first few parts are about building sets of sets of sets... of sets: The first set is S₁ = {a}. The second set is S₂ = P(S) = {0, {a}}. What is the third set S3 = P(S₂) = P(P(S₁))? What is S3|? (2) What is |S₁|? (3) What do you think is the formula for S,? (You might need to use some... in your formula.)

For this Big Problem, we're going to investigate induction in different settings, and think about when it does and doesn't work. (1) The first few parts are about building sets of sets of sets... of sets: The first set is S₁ = {a}. The second set is S₂ = P(S) = {0, {a}}. What is the third set S3 = P(S₂) = P(P(S₁))? What is S3|? (2) What is |S₁|? (3) What do you think is the formula for S,? (You might need to use some... in your formula.)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.3: Divisibility
Problem 49E: 49. a. The binomial coefficients are defined in Exercise of Section. Use induction on to prove...
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For this Big Problem, we're going to investigate induction in different settings, and think about when it does and doesn't work. (1) The first few parts are about building sets of sets of sets... of sets: The first set is S₁ {a}. The second set is S₂ P(S) = {0, {a}}. What the third set S3 = P(S₂) = P(P(S₁))? What is S3|? = (2) What is |S₁|? = (3) What do you think is the formula for S? (You might need to use some... in your formula.) (4) Prove that formula by induction. (5) This part is about watching TV. Suppose you are planning to watch episodes of your favorite TV show over the weekend and also do some homework. Each episode is twenty minutes long. Do you think it's okay to watch one episode? (6) If you've already watched some TV, twenty minutes more to watch one more episode won't make much difference, right? (7) Use the answers to 5 and 6 to prove by induction that it's okay to watch all the episodes over the weekend. (8) Sometimes mathematicians think that everything in life can be turned into a math problem, but that's not quite true. Sometimes using logic isn't useful in the "real world". For the TV question, did you think induction was useful or not? Why?
Transcribed Image Text:For this Big Problem, we're going to investigate induction in different settings, and think about when it does and doesn't work. (1) The first few parts are about building sets of sets of sets... of sets: The first set is S₁ {a}. The second set is S₂ P(S) = {0, {a}}. What the third set S3 = P(S₂) = P(P(S₁))? What is S3|? = (2) What is |S₁|? = (3) What do you think is the formula for S? (You might need to use some... in your formula.) (4) Prove that formula by induction. (5) This part is about watching TV. Suppose you are planning to watch episodes of your favorite TV show over the weekend and also do some homework. Each episode is twenty minutes long. Do you think it's okay to watch one episode? (6) If you've already watched some TV, twenty minutes more to watch one more episode won't make much difference, right? (7) Use the answers to 5 and 6 to prove by induction that it's okay to watch all the episodes over the weekend. (8) Sometimes mathematicians think that everything in life can be turned into a math problem, but that's not quite true. Sometimes using logic isn't useful in the "real world". For the TV question, did you think induction was useful or not? Why?
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(4) Prove that formula by induction. (5) This part is about watching TV. Suppose you are planning to watch episodes of your favorite TV show over the weekend and also do some homework. Each episode is twenty minutes long. Do you think it's okay to watch one episode? (6) If you've already watched some TV, twenty minutes more to watch one more episode won't make much difference, right?
Transcribed Image Text:(4) Prove that formula by induction. (5) This part is about watching TV. Suppose you are planning to watch episodes of your favorite TV show over the weekend and also do some homework. Each episode is twenty minutes long. Do you think it's okay to watch one episode? (6) If you've already watched some TV, twenty minutes more to watch one more episode won't make much difference, right?
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