GO Classes Test Series 2024 | Mock GATE | Test 14 | Question: 32

Question

Which of the following sets has the greatest cardinality?

• A. The set of real numbers R
• B. The set of all functions from R to {0,1}
• C. The set of all finite subsets of natural numbers
• D. The set of all finite-length binary strings

Answer 1

Let's compare the cardinalities of the given sets:

1. The set of real numbers R has the cardinality of the continuum, which is uncountably infinite and has the same cardinality as the points on a line.

2. The set of all functions from R to {0,1} has a cardinality strictly greater than the cardinality of the continuum. This is because there are 2^c (where c is the cardinality of R) different functions from R to {0,1}. In other words, it has a larger cardinality than the set of real numbers.

3. The set of all finite subsets of natural numbers has a countably infinite cardinality, which is the same as the cardinality of the natural numbers themselves.

4. The set of all finite-length binary strings also has a countably infinite cardinality. This is because for each positive integer n, there are 2^n different binary strings of length n.

Comparing these, the set of all functions from R to {0,1} has the greatest cardinality among the given sets.

Comments

clear my doubt..

 those which are uncountable infinite must not have cardinality  so 1,2 should be rejected

Considering 3,4

3 is subsets of natural number

4 is numbers only any combination of 1 and 0 in binary will form a number

So subsets must be greater than number themself so c should be answer

Please correct where I am wrong