https://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.Hugoen-UShttps://maximummathematics.com/books/foundational-mathematics/set-theory/sets/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/sets/index.htmlTrying to precisely define a set can be rather tricky. While there are formal definitions and axioms, here we will rely on our intuition. Even without a formal specification, we can still derive many useful results that hold up under scrutiny. In this section, we’ll learn what a set is (intuitively), and how to describe what kinds of things are in a set. Intuitively Defining a Set Sets, or collections, of objects abound in daily life. We could speak of the set of fruits available for purchase at a local grocery store, the set of birds native to North America, the set of components used to build a specific computer, the set of roads from New York to Los Angeles, and so on. We can make note of a couple of things here:https://maximummathematics.com/books/foundational-mathematics/set-theory/size-of-a-set/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/size-of-a-set/index.htmlIn the previous section, we saw sets that can have vastly different numbers of elements. Some sets have a finite number of elements, meaning that if we started to list out all of that set’s elements, we’d eventually be able to stop, having written down every element. In contrast, some sets have infinitely many elements, meaning that if we start listing elements and stop at any point in time, there would still be elements missing.https://maximummathematics.com/books/foundational-mathematics/set-theory/subsets/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/subsets/index.htmlA set can contain a wide variety of objects. They can contain objects that most people interact with on a daily basis — automotive parts that can be used to service a 1967 Camaro, tools used to carve statues out of wood, or art supplies needed to paint a picture. Likewise, they can contain a wide variety of mathematical objects, like numbers, shapes, or axioms. Sometimes, we only care about some of the objects in a set. For example, we may only be interested in automotive parts needed to service a car’s headlights, or we may only be interested in art supplies needed for a fresco painting. Regardless, we’re able to build a lot of structure by constructing new sets from old ones — simply by restricting what elements are included in the new set. Here, we explore this kind of relationship between sets.https://maximummathematics.com/books/foundational-mathematics/set-theory/empty-set/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/empty-set/index.htmlJust because we can put just about anything in a set doesn’t mean we need to have something in a set. There’s a special and unique kind of set that has no elements in it at all! EMPTY SET, NULL SET The empty set is the unique set containing no elements at all. The empty set is sometimes referred to as the null set, and is often symbolized by $\emptyset$.https://maximummathematics.com/books/foundational-mathematics/set-theory/proof-technique-element-arguments/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/proof-technique-element-arguments/index.htmlIn the previous two sections, we provided proofs for two theorems (Theorem 3.4.1 and Theorem 3.3.1). Both proofs required that we select an arbitrary element from some set $A$, and show that because that arbitrarily chosen element satisfies some property, all elements of set $A$ satisfy that property. This is simply a rehash of the concept of Universal Specification and Universal Generalization as discussed in Chapter 2. In this section, we adapt the methods of Universal Specification and Universal Generalization to a new, powerful proof technique we can use for sets.https://maximummathematics.com/books/foundational-mathematics/set-theory/a-set-of-operations-on-sets/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/a-set-of-operations-on-sets/index.htmlWe’ve discussed how to construct new sets by simply taking some of the elements from one set and putting them into a new set. While subsets are vitally important not only in Set Theory but throughout all of mathematics, subsets are formed by only considering one set at a time. We can construct a wide variety of sets by considering two or more sets at a time. In this section, we learn what kinds of sets we can construct by considering more than one set at a time.https://maximummathematics.com/books/foundational-mathematics/set-theory/graphical-depictions-of-sets/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/graphical-depictions-of-sets/index.htmlIn the previous section, we defined the complement, union, intersection, difference, symmetric difference, multiple union, and multiple intersection of sets purely symbolically. While the symbolic definitions are precise, it’s often easier to build an intuition for what these operations actually do by picturing them. The standard way to picture a set is as a circle, with each circle drawn inside a rectangle standing for the universe $\mathcal{U}$. Whenever an operation includes some region formed by these circles, we shade that region green.https://maximummathematics.com/books/foundational-mathematics/set-theory/the-laws-of-set-theory/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/the-laws-of-set-theory/index.htmlUnderstanding the laws of set theory is essential because they solve the problem of determining when two or more sets are equal, through fundamental equations including associativity, commutativity, and distribution. These laws provide a structured framework for defining and manipulating sets, ensuring consistency and precision in operations such as subset, difference, and symmetric difference. In other words, by developing laws of set theory, we can essentially manipulate equations involving sets as if they were algebraic equations. In addition, the laws of set theory will allow us to convert complicated expressions involving sets into simpler ones, much like how we took complicated logical expressions and simplified them into smaller, logically equivalent expressions.https://maximummathematics.com/books/foundational-mathematics/set-theory/proof-technique-exhaustion/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/proof-technique-exhaustion/index.htmlAll of the proof techniques discussed thus far work on sets that have infinitely many elements. For example, we’ve talked about theorems that apply to all even numbers, not merely some of them. For example, for all even numbers, adding one yields an odd number. As another example, no matter which two even numbers are added together, the sum is always another even number. We’ve even talked about theorems that apply to all sets in general, not merely some of them. These include the set operations and the set equalities.https://maximummathematics.com/books/foundational-mathematics/set-theory/set-partitions/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/set-partitions/index.htmlWhen working with a set of related objects, we may want to split that set up into smaller, more manageable sets. For example, we may want to split up the integers based on parity: even integers, and odd integers. When dealing with the real numbers, we may want to split them up into three separate sets: the positive real numbers, the negative real numbers, and the number $0$. One more example may be the positive rational numbers, where we split them up based on how big they are — we may have all of the positive rational numbers less than $1$ in one set, and all of the positive rational numbers greater than or equal to $1$ in the other set.https://maximummathematics.com/books/foundational-mathematics/set-theory/proof-technique-casework/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/proof-technique-casework/index.htmlPreviously, we talked about a proof technique where we examine every single element of a finite set. In a sense, we examined multiple cases, and verified some result for each of those cases. An exhaustive proof is a special kind of proof by casework. Sometimes, we may have trouble demonstrating that some result holds for a set of elements. However, if we partition the set into groups that have their own special traits, and those traits make it easy to show the desired result holds, a proof by casework can be used to establish the desired result for the entire set.https://maximummathematics.com/books/foundational-mathematics/set-theory/russells-paradox/index.htmlMon, 01 Jan 0001 00:00:00 +0000https://maximummathematics.com/books/foundational-mathematics/set-theory/russells-paradox/index.htmlSets are a powerful abstraction that allows us to collect a wide variety of objects into one structure. Typically, all objects within a set share some common characteristic other than mere inclusion in the set. They may be points on the plane, equilateral triangles, even numbers, or even fruits. As powerful as sets are, we haven’t given a formal definition of what a set is. We’ve been relying on an intuitive definition, and though it works well enough for our purposes, sooner or later the cracks start to show. Sometime around 1901, a mathematician named Bertrand Russell formulated his now-infamous paradox that seemed to dismantle the entire theory of sets. The scary thing about the paradox is that a lot of mathematical research and important results rested on the foundations of Set Theory. If Set Theory is wrong, are all results depending on sets wrong as well?