MAS1614 Real Analysis Flashcards

Flashcards covering key concepts from the MAS1614 Real Analysis lecture notes, including logic, proofs, sets, and functions.

Axioms

Mathematical statements that are regarded as true and used as a basis for reasoning.

A ⇒ B (Implication)

A logical connector where "A ⇒ B" is true if either A and B are both true, or if A is not true.

A ⇔ B (Equivalence)

A logical connector that is true if either both A and B are true, or if both A and B are false; often expressed as "A if and only if B."

∀ (For all)

A symbol used to denote "for all" in mathematical statements.

∃ (There exists)

A symbol used to denote "there exists" in mathematical statements.

Direct Proof

Begins with a valid statement from the hypothesis and uses logical deductions to reach the conclusion.

Proof by Contradiction

Begins by assuming the negation of what needs to be proven and proceeds until a contradiction is found.

Induction

A proof technique used with natural numbers, based on showing a statement is true for n = 1 and that if it's true for n, it's also true for n + 1.

Bernoulli’s Inequality

For all n ∈ N and all x ∈ R with x ≥ −1, (1 + x)n ≥ 1 + nx.

Binomial Theorem

A formula expressing (x + y)n as a sum of terms involving binomial coefficients and powers of x and y.

Geometric Sum

For all x ∈ R with x ̸= 1 and all n ∈ N, the sum of xk from k = 0 to n equals (1 − xn+1) / (1 − x).

Set

A collection of objects.

x ∈ A

Denotes that x is an element of set A.

x ∉ A

Denotes that x is not an element of set A.

Union (A ∪ B)

The set containing all elements that are in A or B (or both).

Intersection (A ∩ B)

The set containing all elements that are in both A and B.

Empty Set (∅)

The set that contains no elements.

Disjoint Sets

Sets that have no elements in common (their intersection is the empty set).

A ⊆ B (A is a subset of B)

Every element of A is also an element of B.

Function (f: A → B)

A correspondence that associates each element a in A (the domain) with a unique element b in B.

Range of f (f(A))

The set of all images f(a) for a in A.

Triangle Inequality

For all a, b ∈ R, |a + b| ≤ |a| + |b|.

Injective Function (One-to-one)

A function where f(a) = f(b) implies a = b.

Surjective Function (Onto)

A function where f(A) = B (every element in B has a corresponding element in A).

Bijective Function

A function that is both injective and surjective.

Constant Function

A function where there exists an element b ∈ B such that f(a) = b for all a ∈ A.

Identity Function (idA)

A function where f(a) = a for all a ∈ A.

Composition of Functions (g ◦ f)

Given f : A → B and g : B → C, the function (g ◦ f)(a) = g(f(a)).

Inverse of f (f −1)

If f is bijective, the unique function g : B → A such that f ◦ g = idB and g ◦ f = idA.