IIT Madras MS written test 2019

Question

Which of the following infinite sets have the same cardinality?
$\mathbb{N}$ : Set of Natural numbers

$\mathbb{E}$ : Set of Even numbers

$\mathbb{Q}$ : Set of Rational numbers

$\mathbb{R}$ : Set of Real numbers

 

• A. $\mathbb{N}$ and $\mathbb{E}$
• B. $\mathbb{Q}$ and $\mathbb{R}$
• C. $\mathbb{R}$ and $\mathbb{N}$
• D. None of the above

Comments

Ans will be $D)$

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But we can define a one-to-one and onto mapping between N and E, right? Like this:
1 -> 2
2 -> 4
3 -> 6
4 -> 8
.
.
.

(this goes on up to infinity from both sides)
So can option A be correct?

http://www.maths.nuigalway.ie/~rquinlan/MA180calculus/section2-3.pdf On this link, it has been explained that if we can find a bijective function between 2 sets, no matter whether they're finite or infinite, then those 2 sets have the same cardinality. This way, the set of natural numbers and the set of integers have the same cardinality, so, even the set of natural numbers and the set of even numbers should have the same cardinality.

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E is subset of N

Both cannot be same. Moreover even number can be -ve too. But Natural number cannot be -ve

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@SPluto

N, E, Q  all are countably infinite sets.. right? 

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@ankitgupta.1729

all will be countable here. Nothing like power set. So, no question of uncountale set

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only N, E, Q are countably infinite sets.. R is uncountably infinite set.. based on cardinality, sets either have finite size, countably infinite(whose cardinality is denoted by aleph null) and uncountably infinite (whose cardinality is denoted by $\epsilon$)... since first 3 are countably infinite sets, so all these 3 should have same cardinality aleph null.. set of real numbers is uncountably infinite set whose cardinality is more than aleph null..

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yes, u r right.

Subset of natural number is countable.So, Q and R both will be uncountable

https://en.wikipedia.org/wiki/Countable_set

but here asking about infinite set. So, all are infinite. And cardinality of all be different, because all are different in type.

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@ankitgupta.1729

countably infinite set must be a bijective function

right? https://gateoverflow.in/216802/set-theory

Answer 1

it should be D . bcoz every 1 is differ in definition.