For each of the following sets, is it countable or uncountable? Briefly explain why. Your explanation should be something like: "There is a surjective function from Z² to the set (with your function written explicitly). Since Z² is countable, the set is countable. 1. R XZ 2. Z[i] = {a + bi | a, b = Z²} 3. {| p is a prime number}. NT 4. {x ER | x = or x = 2 ne 3 for some ne Z}. 5. M₂ (R), the set of 2 × 2 matrices over R. 6. S = {Straight lines in R²}.
For each of the following sets, is it countable or uncountable? Briefly explain why. Your explanation should be something like: "There is a surjective function from Z² to the set (with your function written explicitly). Since Z² is countable, the set is countable. 1. R XZ 2. Z[i] = {a + bi | a, b = Z²} 3. {| p is a prime number}. NT 4. {x ER | x = or x = 2 ne 3 for some ne Z}. 5. M₂ (R), the set of 2 × 2 matrices over R. 6. S = {Straight lines in R²}.
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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Question
![For each of the following sets, is it countable or uncountable? Briefly explain
why. Your explanation should be something like: “There is a surjective function
from Z² to the set (with your function written explicitly). Since Z2 is countable,
the set is countable.”
1. R XZ
2. Z[i] = {a + bi | a, b = Z²}
3. {| p is a prime number}.
Nπ
4. {x € R | x = or x =
2
ne
3
for some n € Z}.
5. M₂(R), the set of 2 × 2 matrices over R.
6. S {Straight lines in R²}.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F382d69cf-ffe2-43c0-99c4-21bedf550518%2Fe1d4713a-7b3e-4075-86a1-5b77828c0667%2F8114y9_processed.png&w=3840&q=75)
Transcribed Image Text:For each of the following sets, is it countable or uncountable? Briefly explain
why. Your explanation should be something like: “There is a surjective function
from Z² to the set (with your function written explicitly). Since Z2 is countable,
the set is countable.”
1. R XZ
2. Z[i] = {a + bi | a, b = Z²}
3. {| p is a prime number}.
Nπ
4. {x € R | x = or x =
2
ne
3
for some n € Z}.
5. M₂(R), the set of 2 × 2 matrices over R.
6. S {Straight lines in R²}.
=
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