Analysis

MATH 131A - Analysis

Complex Analysis for Applications

MATH 132 - Complex Analysis for Applications

<p>(Formerly numbered 95.) Lecture, three hours; discussion, one hour. Enforced requisite: course 32A. Not open for credit to students with credit for course 131A or 132. Introduction to writing and evaluating rigorous mathematical proofs. Covers basic logic; structure of mathematical proofs; sets, functions; natural numbers and induction; axiomatic systems; orderings and equivalence relations; topology of the real numbers; sequences and convergence. P/NP or letter grading.</p>

Mathematics

College of Letters and Science

Container 1

MATH 11 - Transition to Upper Division Mathematics

Transition to Upper Division Mathematics

Reason about the quotient of a set modulo an equivalence relation, and whether or not a given function on such a set is well defined.

Prove statements about natural numbers by mathematical induction.

Prove theorems involving basic set theory, including concepts such as unions, inter-sections, complements, functions, injectivity, surjectivity, and inverse functions.

Translate mathematical statements in English into logical expressions, and  translate such formulas into statements in English.

Prove statements of more complicated forms that combine multiple of the above concepts.

Apply the rules of boolean algebra to logical expressions, to transform them into equivalent forms.

Take an arbitrarily complicated definition, which may include multiple quantifiers (for all, there exists) and logical connectors (and, or, implies) and construct its logical negation.

Prove statements of the form “𝑃 or 𝑄”.

Prove statements of the form “For all 𝑥 ∈ 𝑋, 𝑃 (𝑥)”.

Prove statements of the form “There exists 𝑥 ∈ 𝑋 such that 𝑃 (𝑥)”.

Prove theorems in the context of an axiomatic system, i.e. such that the proof follows only from a given set of axioms and other theorems derived from those axioms.

Prove statements of the form “𝑃 if and only if 𝑄”.

Prove basic theorems from real analysis by combining many of the aforementioned skills.

Analyze definitions from real analysis, with multiple nested quantifiers, to determine their meaning.

Evaluate a proof to find any logical errors or other flaws in the way it is written.

Prove statements of the form “If 𝑃 then 𝑄”, either by direct proof, by contrapositive, or by contradiction.


General Catalog

Transition to Upper Division Mathematics

MATH 11

4 units