https://maximummathematics.com/books/foundational-mathematics/index.htmlThe first book in the Maximum Mathematics series.Hugoen-UShttps://maximummathematics.com/books/foundational-mathematics/table-of-contents/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/table-of-contents/index.htmlLogic Propositions Modeling Logic with Truth Tables Logical Order of Operations Satisfiability Laws of Logic Simplifying Logical Expressions Application: Switching Networks Open Propositions Quantifiers Quantified Laws of Logic Multiple Quantifiers Application: Modeling Logic Puzzles Proof Set Theory Ubiquitous Sets of Numbers Functions Coordinate Systems Functions on Real Numbers Implicitly Defined Functions Parametric Functions Sequences Basic Counting Techniques Relations Graphs Algorithms 3D Coordinate Systems 3D Functions Linear Systems Inequalities Trigonometry Complex Numbershttps://maximummathematics.com/books/foundational-mathematics/logic/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/logic/index.htmlIn most mathematics courses a student has taken so far, they have usually been presented with a collection of results — facts they can use whenever warranted. Polynomials, for instance, come with a wide variety of such facts: how to divide one polynomial by another, how to find the roots of a quadratic equation, and so on. Geometry offers just as many. The Angle Bisector Theorem tells us that the bisector of an angle in a triangle divides the opposite side into two segments proportional to the triangle’s other two sides. A wide variety of theorems describe the chords of a circle. And the famous Pythagorean Theorem relates the three sides of a right triangle.https://maximummathematics.com/books/foundational-mathematics/proof/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/proof/index.htmlIn the previous chapter, we built up the tools of propositional and quantified logic: propositions, connectives, truth tables, the laws of logic, and quantifiers. Along the way, we occasionally ran into the implication — one proposition claiming that another must follow from it — without stopping to give it the attention it deserves. That attention is where we begin this chapter. From there, we turn those tools toward their real purpose: building arguments whose conclusions are guaranteed to be true, and proving that mathematical statements — not just isolated propositions, but general claims about numbers, shapes, and structures — are true beyond any doubt.https://maximummathematics.com/books/foundational-mathematics/set-theory/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/index.htmlThe mathematical logic that we’ve studied in the previous two chapters is foundational to all types of math. However, as demonstrated in the last few sections of Chapter 2, mathematicians rarely lay out all of the full, gory details when writing proofs. Instead, they rely on axioms, definitions, and previous theorems to work out the desired result. Occasionally, propositional logic may be used when doubts arise about the validity of a given proof, but that too is rare.