Discrete Mathematics #03 | Set Theoretical Operations
In the previous lectures, we established the foundations of sets and how to represent them mathematically via Venn diagrams and formal notation. Several set-theoretical operations are considered essential, and thus, we will discuss them.
The empty set
The empty set (denoted by ∅ or {}) represents the set that has no elements.
Clearly ∅ is a subset of every set.
The singleton
It is a set that contains just one element x. We denote it by A = {x}.
Subset
A set S is a subset of a set T (denoted S ⊆ T) if whenever s ∈ S then s ∈ T, and in this case, the set T is said to be a superset of S (denoted T ⊇ S). For instance: {1, 2} ⊆ {1, 2, 3}
A set S is a proper subset of a set T (denoted S ⊂ T) if S ⊆ T and S ≠ T. That is, every element of S is an element of T and there is at least one element in T that is not an element of S. In this case, T is a proper superset of S (denoted T ⊃ S).
Cartesian Product
The Cartesian product allows a new set to be created from existing sets. The Cartesian product of two sets S and T (denoted S × T) is the set of ordered pair {(s,t) | s ∈ S, t ∈ T}. In other words, the set of all ordered pairs (s, t) such that s belongs to S and t belongs to T. Note that the Cartesian product is not commutative, which means S × T != T× S. For example:
S = { 1, 2 }
T = { 3, 5}
S × T = { (1,3), (1,5), (2, 3), (2,5) }
T × S = { (3,1), (3,2), (5,1), (5,2) }
Therefore: S × T ≠ T × S. This implies that two ordered pairs (s1, t1) and (s2, t2) are considered equal if and only if s1 = s2 and t1 = t2.
Power set
The power set of a set A denotes the set of subsets of A. For example,
the power set of the set A = {1, 2, 3} has 8 elements and is given by
P(A) = { ∅, {1}, {2}, {3}, {1,2}, {1, 3}, {2, 3}, {1, 2, 3} }
There are 2 ^ 3 = 8 elements in the total power of A = {1, 2, 3}.
Set Difference Operation
The set difference operation A\B yields the elements in A that are not in B. It is defined by A\B = { a | a ∈ A and a ∉ B}.
For A and B defined as A = {1, 2} and B = {2, 3} we have A\B = {1}.
Complement
The complement of a set A (with respect to the universal set U) is the elements in the universal set that are not in A. It is denoted by A′ and is defined as A′ = { u | u a ∈ U and u ∉ A} = U\A. An easier approach is to think of the complement as the all possible values that are not included in set A.
Conclusion
In the previous lecture, we discussed the union and intersection. If you made it that far, that means that you are able to recognize the most iconic set operations. In the next one, we will discuss more the properties of the Theoretical Operations.
