# DM-Sets,Relations and Functions-Q7

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.

reshown Jul 6, 2019

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 (1,710 points)
edited Jul 6, 2019