DM-Sets,Relations and Functions-Q7

+2 votes

Consider an infinite set A. B is a set such that |B| < |A|. There is a bijection from A to Set of Natural numbers N. Which one of the following statements is TRUE about set B?

(A). B can be infinite but not countable
(B). B is finite
(C). B can be uncountable
(D). B cannot be an empty set.

asked Jul 5, 2019 in Discrete Maths by gbeditor (32,710 points)
reshown Jul 6, 2019 by gbeditor

1 Answer

+5 votes

 There is a bijection from A to Set of Natural numbers N

This tells us that A is countable set.

Also given that |B| < |A| 

So we can eliminate option A and C, 

We need to eliminate D too because there is no information which suggests that B cannot be an empty set.

Which leaves us with option \text{B} being correct

EDIT : One of my friend asked why can't B be infinite,

Let's suppose A is set of Whole Numbers and B is set of Natural Numebers

Hence |B| < |A|

No this is incorrect.

Set of Whole numbers and Natural Numbers have same cardinality.

Why? Because the set of natural numbers and the set of whole numbers can be put into one-to-one correspondence with one another. Therefore they have the same cardinality.

So if |B| < |A|  and |A| is \infty then  B is definately finite 

answered Jul 6, 2019 by (1,730 points)
edited Jul 6, 2019 by
what is the answer, B?
you have any doubt?