Cardinality of a countable set can be a finite number. For example, B: {1, 5, 4}, |B| = 3, in this case its termed countably finite or the cardinality of countable set can be infinite. For example, A: {2, 4, 6, 8 …}, in this case its termed countably infinite.Common Traces for Countable Set:
- Cardinality expressed in form
where , ; m may or may not be ∞ - It has finite elements* only in case of countably finite sets.
- It listable in terms of roaster form* an exhaustive list exists which can include every element atleast once, in case of countably infinite list first few elements followed by three dot ellipsis(…).
R : {set of real numbers is uncountable} B : {set of all binary sequences of infinite length}Common Traces for Uncountable Set:
- Cardinality expressed in form
; - It is power set of set with infinite elements
- It is equal set to R set of real numbers
- It is equal set to Q set of irrational numbers
- It is non-listable set
A | B | |
---|---|---|
Countable | Countable | Countable |
Uncountable | Uncountable | Uncountable |
Countable | Uncountable | Uncountable |
P: Set of Rational numbers (positive and negative) Q: Set of functions from {0, 1} to N R: Set of functions from N to {0, 1} S: Set of finite subsets of NWhich of the above sets are countable ? (A) Q and S only (B) P and S only (C) P and R only (D) P, Q and S only Explanation: Please see GATE CS 2018 | Question 58Example-2: Consider the following sets:
S1: Set of all recursively enumerable languages over the alphabet {0, 1}. S2: Set of all syntactically valid C programs. S3: Set of all languages over the alphabet {0, 1}. S4: Set of all non-regular languages over the alphabet {0, 1}.Which of the above sets are uncountable? (A) S1 and S2 (B) S3 and S4 (C) S1 and S4 (D) S2 and S3 Explanation: Please see GATE CS 2019 | Question 43
Recommended Articles