Mathematical Language and Sets

• Mathematics uses a unique language to communicate concepts precisely.
• This language includes specific vocabulary, symbols, and grammatical rules.
• Mathematical language is clear, avoiding ambiguities found in natural languages.
• It is brief, precise, and expresses complex ideas efficiently.

Sets

• A set is a well-defined collection of distinct objects, treated as a single object.
• Elements or members are the objects within a set.
• Sets are represented by uppercase letters (e.g., A, B, C).
• Elements are represented by lowercase letters (e.g., a, b, c).
• "∈" indicates set membership; "a ∈ A" means "a is an element of A."
• "∉" indicates non-membership; "b ∉ A" means "b is not an element of A."
• Sets defined by listing elements within curly braces use the roster method: {1, 2, 3}.
• Element order in a set is irrelevant; duplicates are ignored.
• Sets defined by a property elements must satisfy use set-builder notation: {x | P(x)}, meaning "the set of all x such that P(x) is true."
• The set of all positive integers is {x | x > 0}.

Common Sets

• Natural numbers (ℕ) are {1, 2, 3, ...}.
• Integers (ℤ) are {..., -2, -1, 0, 1, 2, ...}.
• Rational numbers (ℚ) are {p/q | p, q ∈ ℤ, q ≠ 0}.
• Real numbers (ℝ) include all rational and irrational numbers.
• Complex numbers (ℂ) are {a + bi | a, b ∈ ℝ, i is the imaginary unit}.
• The empty set (∅ or {}) contains no elements.

Set Operations

• The union of sets A and B (A ∪ B) contains elements in A, B, or both: A ∪ B = {x | x ∈ A or x ∈ B}.
• The intersection of sets A and B (A ∩ B) contains elements in both A and B: A ∩ B = {x | x ∈ A and x ∈ B}.
• The difference of sets A and B (A - B) contains elements in A but not in B: A - B = {x | x ∈ A and x ∉ B}.
• The complement of set A (A') contains elements not in A, within a universal set U: A' = {x | x ∈ U and x ∈ A}.
• A universal set contains all elements under consideration.

Relations

• A relation involves a set of ordered pairs.
• It describes relationships between elements of two or more sets.
• A binary relation between sets A and B is a subset of the Cartesian product A × B.
• The Cartesian product A × B includes all possible ordered pairs (a, b) where a ∈ A and b ∈ B: A × B = {(a, b) | a ∈ A and b ∈ B}.
• If A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}.
• A relation R from A to B is a subset of A × B.
• If (a, b) ∈ R, 'a is related to b by R', written as aRb.

Types of Relations

• Reflexive: For every a ∈ A, (a, a) ∈ R.
• Symmetric: For every (a, b) ∈ R, (b, a) ∈ R.
• Transitive: For every (a, b) ∈ R and (b, c) ∈ R, (a, c) ∈ R.
• An equivalence relation is reflexive, symmetric, and transitive.

Operations

• Operations are functions that take zero or more inputs (operands) to produce a new value.
• A unary operation takes one operand.
• A binary operation takes two operands.
• Number negation (-x) is a unary operation.
• Addition (+), subtraction (-), multiplication (×), and division (÷) are binary operations.
• A binary operation on a set A is a function from A × A to A.
• The operation takes two elements from A and returns an element also in A.
• Addition is a binary operation on integers (ℤ); adding two integers yields another integer.

Functions

• Functions map each element from the domain to exactly one element in the codomain.
• A function f from A to B, denoted f: A → B, means for every a ∈ A, there's a unique b ∈ B such that (a, b) ∈ f.
• Each input has only one output.
• If f: A → B and (a, b) ∈ f, then f(a) = b.
• 'a' is the function's argument or input, and 'b' is the function's value or output.
• A is the domain of f, and B is the codomain of f.
• The range of f includes all actual output values of f.
• The range is a subset of the codomain.
• Range(f) = {f(a) | a ∈ A}.

Domain and Range

• The domain is all possible input values for which the function is defined.
• For f: A → B, A represents the domain.
• The range includes all actual output values produced from all possible domain inputs.
• The range is a subset of the codomain.
• Finding a function's domain means identifying values that make the function undefined.
• Division by zero must be avoided to keep a function defined.
• Taking the square root of a negative number (in real numbers) must be avoided.
• Taking the logarithm of a non-positive number must be avoided.

Examples of domain and range

• For f(x) = 1/x, the domain is all real numbers except 0 (ℝ - {0}), and the range is also ℝ - {0}.
• For g(x) = √x, the domain is all non-negative real numbers ([0, ∞)), the range is also [0, ∞).
• For h(x) = x^2, the domain is all real numbers (ℝ), and the range is all non-negative real numbers ([0, ∞)).

Function Composition

• The composition of functions f and g, (f ∘ g)(x) = f(g(x)).
• The range of g must be a subset of the domain of f for the composition to be defined.