Mathematical Notations and Terminology

1. Sets

A set is a group of objects represented as a unit.

• Elements: The objects within a set. The symbols $\in$ and $\notin$ denote membership and non-membership.
• Subsets: $A\subseteq B$ means every member of $A$ is also in $B$.
• Proper Subset: $A⊊B$ means $A\subseteq B$ but $A\ne B$.
• Infinite Sets: Sets with infinitely many elements, often denoted with "$\dots$"
  • Natural numbers → $\mathbb{N}=\{1,2,3,\dots \}$
  • Integers → $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$
• Special Sets:
  • Empty Set ($\varnothing$): A set with zero members.
  • Multiset: A set where the number of occurrences of an element matters.
    • $\{7\}\ne \{7,7\}$
• Set Operations:
  • Union ($A\cup B$): Combines all elements from both sets.
  • Intersection ($A\cap B$): Elements common to both sets.
  • Complement ($\bar{A}$): All elements under consideration that are not in A.

Union	Intersection

2. Sequences and Tuples

A sequence is a list of objects in a specific order, designated by parentheses.

• Order and Repetition: Unlike sets, both order and repetition matter in sequences
  • $(7,21,57)\ne (57,7,21)$
• Tuples: Finite sequences. A $k$-tuple has $k$ elements; a 2-tuple is an ordered pair.
• Power Set: The set of all subsets of a given set.

3. Functions and Relations

• Functions: A rule that maps an input (from a domain $D$) to exactly one output (from a range $R$ or codomain).
• Relations: A set of ordered pairs representing a property between elements (e.g., “less than” on integers).
• Equivalence Relation: A special type of relation that is reflexive, symmetric, and transitive.

4. Graphs

A graph consists of a set of nodes (vertices) and edges connecting them.

• Directed vs. Undirected: Directed edges have an orientation $(u\to v)$; undirected edges do not.
• Paths and Cycles: A path is a sequence of nodes connected by edges; a cycle is a path that starts and ends at the same node.
• Connectedness: An undirected graph is connected if every pair of nodes has a path between them.

5. Strings and Languages

• Alphabet ($\Sigma$): Any finite, non-empty set of symbols.
• String: A finite sequence of symbols from an alphabet.
  • Length: The number of symbols in a string.
  • Empty String ($ϵ$): A string of length 0.
• Language: A set of strings.

6. Boolean Logic

A system for manipulating the values TRUE and FALSE (often represented as 1 and 0).

• Operations:
  • Negation ($\neg$): NOT.
  • Conjunction ($\wedge$): AND.
  • Disjunction ($\vee$): OR.
  • Implication ($\to$): IF-THEN.