- Finite Set: A set is called a finite set if it contains a finite number of elements. For example, the set A = {1, 2, 3, 4} is a finite set because it contains 4 elements.
- Infinite Set: A set is called an infinite set if it contains an infinite number of elements. For example, the set B = {1, 2, 3, 4, …} is an infinite set because it contains an infinite number of natural numbers.
- Countably Infinite Set: A set is called countably infinite if it can be put into a one-to-one correspondence with the set of natural numbers. In other words, there exists a bijection from the set of natural numbers to the set. For example, the set C = {1/1, 1/2, 1/3, 1/4, …} is countably infinite because we can match each natural number with a unique element in the set C.
- Countable Set: A set is called countable if it is either finite or countably infinite. For example, the sets A and C are both countable sets because they are either finite or countably infinite.
- Uncountable Set: A set is called uncountable if it is not countable. For example, the set of real numbers R is uncountable because it cannot be put into a one-to-one correspondence with the set of natural numbers.
Some more examples on countably infinite sets:
The set of positive integers Z+ = {1, 2, 3, …} is a countably infinite set because it can be put into a one-to-one correspondence with the set of natural numbers.
- The set of even numbers E = {2, 4, 6, …} is a countably infinite set because it can be put into a one-to-one correspondence with the set of positive integers.
- The set of rational numbers Q = {p/q | p, q ∈ Z, q ≠ 0} is a countably infinite set because it can be put into a one-to-one correspondence with the set of pairs of integers.
- The set of all strings of 0’s and 1’s is a countably infinite set because it can be put into a one-to-one correspondence with the set of natural numbers.
Cardinality: A set S is said to have finite cardinality, denoted |S|, if
the number of distinct elements in S is finite, else the set S is said to
have infinite cardinality.
Operations on sets:
- Union: S ∪ T = {x ∈ U: x ∈ S or x ∈ T}.
- Intersection: S ∩ T = {x ∈ U: x ∈ S and x ∈ T}.
- Complement: Sc = {x ∈ U ∶ x ∉ S}.
- Difference: S\T = {x ∈ U ∶ x ∈ S and x ∉ T}.
- Symmetric difference: S⨁T = S ∪ T \(S ∩ T).
- Cartesian product: S × T = {(x , y) : x ∈ S and y ∈ T} .
- Disjoint sets: S ∩ T = ∅.
Power Set
