Discrete Mathematics Overview

Discrete Mathematics refers to the study of distinct or unconnected elements.
It involves the study of mathematical structures that are fundamentally discrete rather than continuous.

Develops mathematical maturity.
Serves as a gateway to more advanced courses in computer science and mathematics.

Course Organisation:
- 13 sessions from Week 1 (Introduction) to Week 12 (Review and Support Session).
- One online multiple-choice test and one assignment/logbook submission involved.
- The passing mark for the module is 40%.

Set: An unordered collection of distinct objects, referred to as elements or members of the set.
- Example Notation:
- If studenta belongs to Classa, it is denoted as:
  $student<em>a ∈ Class</em>a$

Well-defined: The defining characteristic should be clear and specific.
Collection: Can contain any number of items (including zero).
Cardinality: The number of items in a set is termed the cardinality, denoted by:
$|S|$
Distinct: Repetitions are not allowed and order of elements typically does not matter.
Homogeneous: All members must be of the same type.
Single Entity: Treated as a single variable during computations.

Null Set: A set with cardinality 0, denoted by the Greek letter phi (ϕ), e.g., ϕ = {}.
Singleton Set: A set with cardinality 1, e.g., S = {a}.
Finite Set: A set with a finite number of elements, e.g., S = {a, b, …, z} |S| = 26.
Countably Infinite Set: An infinite set with a discernible pattern, e.g., S = {2, 4, 6, 8, …}.
Uncountably Infinite Set: An infinite set without a specific pattern.
Crisp Set: A set where elements are easily separable.

Venn Diagram:
- Graphical method to represent sets where overlaps represent common members.
- Example: Let A be a set of all birds and B be a set of all flying species. Not all flying species are birds (e.g., mosquitoes).

Membership: Notation and examples provided.
- Example: For a set M = {all mammals}, the complement (not mammals) is denoted as Mᶜ.
Complement: ( U ackslash S ) or the set of elements not in S.

Union: X = A B ; combines elements from both sets A and B.
- Example: If A = {2, 4, 6} and B = {10, 12, 14}, then A ∪ B = {2, 4, 6, 10, 12, 14}.
Intersection: X = A B ; identifies common elements between A and B.
- Example: If A = {2, 6, 8, 10} and B = {6, 9, 10, 12}, then A ∩ B = {6, 10}.
Disjoint Sets: If A B = ϕ , then the sets have no common elements.
Difference: $X = A - B$ ; set contains members from A that are not in B.
- Example: If A = {2, 4, 6, 8} and B = {2, 6, 10, 12}, then A - B = {4, 8}.
Cartesian Product: $X = A imes B$ ; creates all possible ordered pairs.
- Example with subjects: If A = {Physics, Chemistry, Biology} and B = {History, Geography, Sociology}, combinations yield tuples such as (Physics, History), (Chemistry, Geography), etc.
Subsets: A set A is a subset of B (A ⊆ B) if all members of A are in B.
- Example: A = {all prime numbers}, B = {all whole numbers}.
Power Set: The set of all possible subsets of a set S, denoted as P(S).
- Example: For A = {1, 2, 3}, P(A) = {ϕ, 1, 2, 3, (1,2), (1,3), (2,3), {1,2,3}}.
- Theorem: $|P(S)| = 2^{|S|}$ \n

For academic questions, students should:
1. Raise hands during lectures.
2. Use the VLE forum for questions.
3. Email the teaching team using the specified formatting.

Module Leader: Aarbaz Alam (alema13@lsbu.ac.uk)
Tutors: Aarbaz Alam, Dr. Bugra Alkan, Dr. Louis Spring, Paul Carden, Julie McCarthy, Bisi Bode Kolawole, Louie Webb.

Study lectures actively.
Solve tutorial problems regularly.
Review reference materials such as Kenneth H. Rosen's "Discrete Mathematics and Its Applications" (Eighth Edition).

Attend lectures punctually and ensure to sign the attendance sheet multiple times as required.

Primary textbook: Kenenth H. Rosen - "Discrete Mathematics and Its Applications" (Eighth Edition).