Numbers and logic¶
Part 1 of the lecture notes. Sets, logic, real numbers and suprema, summations, induction, complex numbers.
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1. Sets
Informal introduction to set theory · Number sets · Relations between sets · Operations on sets · …
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2. Basics of logic and proof techniques
Logical symbols · Universal implications and proofs · Law of contraposition · Sufficient conditions and necessary conditions · …
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3. Number sets and intervals
The five main number sets · Intervals
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4. Binary relations
Binary relations · Partial order relations · Total order relations · Functions
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5. Ordered fields, supremum/infimum and the completeness axiom
Properties … and … · Commensurable magnitudes · Geometric representation of … · Total order relations · …
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6. Roots, powers, logarithms and modular arithmetic
Roots, powers, logarithms · Modular arithmetic
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7. Summations and geometric progressions
Summations · Geometric progressions
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8. Principle of mathematical induction
The principle of mathematical induction · Proofs based on the principle of mathematical induction
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9. Factorials, binomial coefficients and triangle inequality
Factorials · Binomial coefficients · Absolute value
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10. The Fibonacci sequence
The Fibonacci sequence
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11. Cardinality of infinite sets
Countable cardinality · Cardinality of the continuum
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12. Complex numbers
Definition of … and field structure · Algebraic form of complex numbers · Trigonometric form of complex numbers · Quadratic equations