Math topics
Chapter and section references are from Corbae, Stinchcombe and Juraj (2009), “An Introduction to Mathematical Analysis for Economic Theory and Econometrics.'' I only recommend you follow Corbae et al. if you are already comfortable with highly-technical math. Otherwise, use this list of topics as guidance while using one of the recommended sources on the Textbooks page and following the guidance on the Math preparation page.
Logic
Chapter 1 in Corbae, Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Statements, Sets, Subsets, Implication | Section 1.1 | X | X | X |
| Ands, Ors, Nots | Section 1.2.a | X | X | X |
| Implies, Equivalence | Section 1.2.b | X | X | X |
| Vacuous Statements | Section 1.2.c | X | X | X |
| Indicators | Section 1.2.d | X | X | |
| Logical Quantifiers | Section 1.4 | X | X | |
| Taxonomy of Proofs | Section 1.5 | X | X | X |
Set Theory
Chapter 2 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Notation for sets | 2.2.2, top pg. 21 | X | X | |
| Useful theorems on sets | 2.2.4, 2.2.6 | X? | ||
| Cartesian Product | 2.3.1 | X | X | X |
| Relation | 2.3.4, 2.3.5 | X | ||
| Function | 2.3.8 | X | X | X |
| Correspondence | 2.3.12 | X | X | X |
| Image | 2.3.16, 2.6.1, 2.6.4 | X | X | |
| Cardinality | 2.3.17 | X | X | X |
| Equivalence Class | 2.4.1, 2.4.5 | X | X | X |
| Partition | 2.4.9 | X | X? | |
| Inverse, Inverse Image | 2.6.7, 2.6.10, 2.6.13 | X | X | X |
| Level Sets of Functions | 2.6.12 | X | X | X |
| One-to-One / Injection | 2.6.15 | X | X | |
| Onto / Surjection / Bijection | 2.6.17 | X | X | |
| Composite Functions | 2.6.20, 2.6.23, 2.6.26 | X | X |
The Space of Real Numbers
Chapter 3 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| The `Why' | Section 3.1 and 3.10 | X | ||
| Algebraic Properties of $\mathbb{Q}$ | 3.2.3 | X | X | |
| Distance in $\mathbb{Q}$ | 3.3.1, 3.3.2 | X | X | |
| Sequence | 3.3.3 | X | X | X |
| Subsequence | 3.3.5 | X | X | |
| Cauchy | 3.3.7, 3.4.8 | X | ||
| Bounded | 3.3.12, 3.3.13, 3.6.1 | X | X | X |
| Real Numbers | 3.3.19 | X | ||
| Algebraic Properties of $\mathbb{R}$ | 3.3.23 | X | X | |
| Distance in $\mathbb{R}$ | 3.4.1, 3.4.2, 3.4.3 | X | X | |
| Convergence | 3.4.9, 3.4.10, 3.4.15 | X | X | X |
| Completeness of $\mathbb{R}$ | 3.4.16 | X | ||
| Supremum / Infimum | 3.6.2, 3.6.5 | X | X | X |
The Finite-Dimensional Metric Space of Real Vectors
Chapter 4 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Metric Space | 4.1.1 | X | ||
| Convergence, Limit | 4.1.4 | X | X | X |
| Complete | 4.1.6, 4.4.5 | X | X | |
| Open ball | 4.1.9 | X | ||
| Open | 4.1.10, 4.1.11 | X | X | |
| Open neighborhood | 4.1.12 | X | ||
| Open cover | 4.1.18 | X | ||
| Compact | 4.1.19, 4.7.15 | X | X | |
| Connected | 4.1.21 | X | X | |
| Continuous | 4.1.22, 4.7.20, 4.85 | X | X | X |
| Vector Space | 4.3.1 | X? | X | |
| Normed Vector Space | 4.3.7 | X? | ||
| Inner / Dot Product | 4.3.9 | X | X | X |
| Cauchy-Schwartz Inequality | 4.3.10 | X | ||
| \textit{p}-Norms | Section 4.3.c | X? | X | |
| Characterizing Closed Sets | Section 4.5.a | X | X | |
| Closure of a Set | 4.5.4 | X | ||
| Boundary of a Set | 4.5.5 | X | X | |
| Accumulation / Cluster / Limit Point | 4.5.7 | X | X | |
| Closure and Completeness | 4.5.12, 4.5.13 | X | X | |
| Bounded | 4.7.8 | X | X | X |
| Applications of Compactness | Section 4.7.f | X | X | X |
| Basic Existence Result | 4.8.11, 4.8.16 | X | X | |
| Upper Hemicontinuity | 4.10.20 | X | X | |
| Theorem of the Maximum | 4.10.22, 6.1.31 | X | X | |
| Upper Semicontinuity | 4.10.29 | X | X? | |
| Connected | 4.1.12, 4.12.3, 4.12.4 | X | X | X |
| Interval | 4.12.1, 4.12.2 | X | X | X |
| Intermediate Value Theorem | 4.12.5 | X | X | X |
Finite-Dimensional Convex Analysis
Chapter 5 in Stinchcombe and Juraj (2009)
| Topic | Reference | Background | Proofs | Direct/General |
|---|---|---|---|---|
| Convex Combination | 5.1.2 | X | X | X |
| Convex Preferences and Technologies | Section 5.1.c | X | ||
| Returns to Scale | 5.1.13 | X | ||
| Convex Hull | 5.4.6 | X | ||
| Upper Contour Set | 5.4.23 | X | X | X |
| Affine Combination | 5.6.16 | X | ||
| Interior | 5.5.1, 5.5.2 | X | ||
| Concave Function | 5.6.1, 5.6.2 | X | X | X |
| Affine Function | 5.6.6 | X | X | |
| Quasi-Concave | 5.6.12 | X | X | X |
| Single-Peaked | 5.6.13 | X | ||
| Implicit Function Theorem | Sections 2.8.a and 5.9.b | X | X | |
| Envelope Theorem | Section 5.9.c | X | X |
Sections 5.8 - 5.10 contain results on optimization. The facts and results that you need should already be familiar from math camp so I do not list them separately.