Mathematics
Mathematics has developed through the close interconnection of many different fields. WIIS provides learning materials for the systematic study of university-level mathematics, covering topics from logic, set theory, and real analysis to calculus, linear algebra, topology, convex analysis, and probability theory. By carefully building up definitions, theorems, and proofs, we emphasize not only individual results, but also the ideas behind them and the connections between different areas of mathematics.
Fields of Mathematics
Logic provides the foundation for mathematical proof and reasoning. Through propositional logic and predicate logic, WIIS systematically explains logical formulas, truth values, quantifiers, and the basic ideas behind proof. These materials provide a starting point for studying university-level mathematics and other theoretical disciplines.
Set theory is the common language of modern mathematics. Focusing on sets, functions, relations, cardinality, and order relations, this course systematically explains the fundamental concepts needed to understand a wide range of mathematical fields. It provides a foundation for further study in areas such as real analysis, linear algebra, and topology.
Probability theory is a mathematical framework for analyzing uncertain phenomena. It provides a systematic study of probability spaces, random variables, discrete and continuous probability distributions, and asymptotic theory, laying the foundation for further study in statistics and machine learning.
New Materials
Replacing a given formula with another formula that is logically equivalent to it is called an equivalence transformation. In predicate logic as well, the binary relation expressing logical equivalence is an equivalence relation, since it satisfies the reflexive, symmetric, and transitive laws.
For formulas \(A\) and \(B\), if \(A\leftrightarrow B\) is a tautology, that is, if \(A\leftrightarrow B\) has the value \(1\) under every interpretation, then \(A\) and \(B\) are said to be necessary and sufficient conditions for each other.
If the implication \(A\rightarrow B\) involving formulas \(A\) and \(B\) is a tautology, that is, if the proposition obtained from \(A\rightarrow B\) is true under every interpretation, then \(B\) is said to be a necessary condition for \(A\), and \(A\) is said to be a sufficient condition for \(B\).
Union events are measurable. The additivity of the probability measure is useful for determining the probability of the union of mutually disjoint events, while subadditivity and the addition theorem are useful for unions of events that are not necessarily mutually disjoint.
The empty event is measurable, and its probability is zero. We derive these properties from the axioms of probability.
We explain the definition of a probability space by clarifying the relationships among the sample space, family of events, sigma-algebra, and probability measure. We also examine the differences between finite, countable, and uncountable sample spaces and illustrate the construction of a general probability space using infinite coin tossing as an example.