DM - Functions & Relations -Q6

+1 vote
Two sets X and Y are said to have the same cardinality if and only if there is a one-to-one correspondence(i.e. Bijection) from X to Y. When A and B have the same cardinality, we write |X|=|Y|.

Consider an infinite set A. B is a set such that |B| < |A| (i.e. There is no bijection possible between A and B; and B has less cardinality than A). There is a bijection between A and 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 may be infinite.
asked Jun 24 in Discrete Maths by gbeditor (44,500 points)
edited 3 days ago by deepak-gatebook

1 Answer

0 votes
 
Best answer

 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, 

Which leaves us with option \text{B},D. 

NOTE : Why can't Set 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 countable infinite And there is No bijection from B to A then  B is definately finite 

answered 4 days ago by deepak-gatebook (112,390 points)
edited 3 days ago by deepak-gatebook
Sir i have one confusion here ..
Lets say A is set of all natural numbers . And B is set of all even numbers ?
Both are infinite but still B will have less elements than A then how are we comparing them?

Are we here only seeing the possibility that option A and C can be easily eliminated and D is also not sure , but B might be possible ? Is this also true that in some cases B can be false?
/// Lets say A is set of all natural numbers . And B is set of all even numbers ?
Both are infinite but still B will have less elements than A then how are we comparing them? ////

Both sets "Set of natural numbers" and "set of even numbers" have SAME Cardinality because there is a bijection possible from one set to other.

Cardinality can be defined as "number of elements in the set"  only for FINITE sets. For ALL sets(finite or infinite), Cardinality is defined to the Bijection between two sets and then we say that if there is a bijection from one set to other then they have same cardinality.

Read Kenethe Rosen(Set Chapter, Cardinality Topic)

For infinite sets the definition of cardinality provides a relative measure of the sizes of two sets, rather than a measure of the size of one particular set.
Answer:
...