A set is a collection of objects called elements of the set. Why not use the word “collection” and eliminate the word “set”, thereby having fewer words to worry about? “Collection” is a common word whose generic meaning is understood by most people. The use of the word “set” means that there is also a method to determine whether or not a particular object belongs in the set. We then say that the set is well-defined. For example, it is easy to decide that the number 8 is not in the set consisting of the integers 1 through 5. After all, there are only five objects to consider and it is clear that 8 is not one of them by simply checking all five.

A basic problem here is now to indicate sets on paper and verbally. As seen above, a set could be described with a phrase such as “the integers 1 through 5” and the speaker hopes that it is understood. Symbollically, we use two common methods to write sets. The roster notation is a complete or implied listing of all the elements of the set. So and are examples of roster notation defining sets with 4 and 20 elements respectively. The ellipsis, “ ”, is used to mean you fill in the missing elements in the obvious manner or pattern, as there are too many to actually list out on paper. The set-builder notation is used when the roster method is cumbersome or impossible. The set B above could be described by . The vertical bar, “|”, is read as “such that” so this notation is read aloud as “the set of x such that x is between 2 and 40 (inclusive) and x is even.” (Sometimes a colon is used instead of |.) In set-builder notation, whatever comes after the bar describes the rule for determining whether or not an object is in the set. For the set the roster notation would be impossible since there are too many reals to actually list out, explicitly or implicitly.

To discuss and manipulate sets we need a short list of symbols commonly used in print. We start with five symbols summarized in the following table.

The first symbol, , indicates membership of an object in a particular set. The negation of this, or nonmembership is often indicated by “ ” (“x is not in A”). The subset relation, , states that every element of A is also an element of B. Logically, this would be: if then The union and intersection operators form new sets by the following rules. The set is defined to be while is defined to be . Finally, the complement of a set consists of those objects that are not in the given set. This presents a minor problem. If then clearly I am not in A so should I be considered an element of ? Normally not, I think. Underlying a discussion or argument involving sets is usually a large set called the universal set or universe of the discourse and is commonly denoted by U. This universe may be implied or stated explicitly. Operations involving union, intersection or complement are understood to be contained in this universe. For example, if we were discussing real numbers (so that our universe would be the set of reals) and mentioned the set A above with 3 elements, it is understood that consists of those real numbers not in A. This conveniently excludes me from the set .

Arbitrary unions and intersections

The notions of union and intersection can be easily extended to more than two sets.

For finitely many sets, say , we write or and or . If we have a sequence of sets we write and
Perhaps the most general form of this notation is a collection of sets with subscripts in some general index set , . Here could be finite or infinite. We then write and . So an alternate way to write would be where N is the set of natural numbers.
Exactly what these symbols mean is as follows.
By we mean and by we mean
Check that these become our original definitions of set union and intersection when the indexing set has just two elements, and then we can just call the two sets A and B. An example of a large indexing set would be the set of reals R. If then would be the entire Cartesian plane written as a union of all vertical lines in the plane.

Some care must be taken when dealing with negations involving union and intersections. Since means there is at least one so that , then means x is not in any . (Logically we have ) Similarly, Since means for every , then means there is at least one with . (Logically we have )

Manipulating Unions and Intersections

The important rules and are known as DeMorgan’s Laws. These rules also extend to arbitrary unions and intersections as and

In manipulating expressions involving set operations it is often convenient to distribute one set operation over another. Specifically we have and

1. The symmetry group of the tetrahedron S(T).

   To calculate the order of the group, oberve that a given vertex can be moved to one of four positions. There is a choice of three for a second and two for a third. Hence |S(T)| = 24.
   Any symmetry determines a permutation of the four vertices so we get a map : S(T) S₄ to the Symmetric group which is easily seen to be an isomorphism.
   Since a transposition (swapping a pair of vertices) corresponds to a reflection (an opposite symmetry), the subgroup S_d(T) of direct symmetries corresponds to the Alternating subgroup A₄.
   We will see a different way of thinking about this group later.

All the other Platonic solids are symmetric about their centres and so (See Exercises 4 Question 5) the full group of symmetries S(X) is isomorphic to the direct product S_d(X) < J > where S_d(X) is the subgroup of direct symmetries and < J > is the subgroup generated by the map x -x.

1. The symmetry group of the cube or octahedron S(C).

   Because these two solids are dual to each other they have the same symmetry group.
   Arguing as in the last case, the order of the group of direct symmetries (all rotations) is |S_d(C)| = 8 3 = 24.
   The elements are:
   3 rotations (by p/2 or p) about centres of 3 pairs of opposite faces. [9]
   1 rotation (by p) about centres of 6 pairs of opposite edges. [6]
   2 rotations (by 2p/3) about 4 pairs of opposite vertices (diagonals). [8]
   Together with the identity this accounts for all 24 elements.

   The orders of these elements suggests the SL_d(C) S₄. In fact every rotation determines a permutation of the four diagonals and this defines the isomorphism.

   Hence S_d(C) S₄ and S(C) S₄ < J > with order 48.

    

2. The symmetry group of the dodecahedron or icosahedron S(D).

   Because these two solids are dual to each other they have the same symmetry group.
   Arguing as before, the order of the group of direct symmetries (all rotations) is |S_d(D)| = 20 3 = 60.
   The elements are:
   4 rotations (by multiples of 2p/5) about centres of 6 pairs of opposite faces. [24]
   1 rotation (by p) about centres of 15 pairs of opposite edges. [15]
   2 rotations (by 2p/3) about 10 pairs of opposite vertices. [20]
   Together with the identity this accounts for all 60 elements.

   This suggests that S_d(D) A₅ which has 24 5-cycles, 20 3-cycles and 15 permutations of the shape (..)(..).

   In fact one can embed five cubes in the dodecahedron which are permuted by each rotation. Alternatively, one may embed five tetrahedra (partitioning the 20 vertices) and these are permuted also.

   Hence S_d(D) A₅ and S(D) A₅ < J > with order 120.

   Remark

   It is tempting to believe that the full symmetry group S(D) is actually isomorphic to S₅ but one can check that the reflections in S(D) lead to even permutations of the tetrahedra and so the full symmetry group is not S₅.

• Considering the natural numbers as a subset of the real numbers, and assuming that we know already that the real numbers are complete (again, either as an axiom or a theorem about the real number system), i.e., every bounded (from below) set has an infimum, then also every set A of natural numbers has an infimum, say a^*. We can now find an integer n^* such that a^* lies in the half-open interval (n^*-1, n^* ], and can then show that we must have a^* = n^*, and n^* in A.
• In axiomatic set theory, the natural numbers are defined as the smallest inductive set (i.e., set containing 0 and closed under the successor operation). One can (even without invoking the regularity axiom) show that the set of all natural numbers n such that “{0,…, n} is well-ordered” is inductive, and must therefore contain all natural numbers; from this property it is easy to conclude that the set of all natural numbers is also well-ordered.

In the second sense, the phrase is used when that proposition is relied on for the purpose of justifying proofs that take the following form: to prove that every natural number belongs to a specified set S, assume the contrary and infer the existence of a (non-zero) smallest counterexample. Then show either that there must be a still smaller counterexample or that the smallest counterexample is not a counter example, producing a contradiction. This mode of argument bears the same relation to proof by mathematical induction that “If not B then not A” (the style of modus tollens) bears to “If A then B” (the style of modus ponens). It is known light-heartedly as the “minimal criminal” method and is similar in its nature to Fermat’s method of “infinite descent“.

Garrett Birkhoff and Saunders MacLane wrote in A Survey of Modern Algebra that this property, like the least upper bound axiom for real numbers, is non-algebraic — i.e., it cannot be deduced from the algebraic properties of the integers (which form an ordered integral domain). It therefore characterizes integers among integral domains; every well-ordered integral domain is isomorphic to the integers.

Well-Ordering Principle

In this topic, we first state the Well-Ordering Principle and then we use it to prove the Archimedean Property and the Principal of Mathematical Induction. We then prove that the Well-Ordering Principle and the Principle Mathematical Induction are equivalent statements.

Definition (Well-Ordering Principle) The Well-Ordering Principle is the following statement:

every nonempty set of positive integers contains a least element.
The Principle of Well-Ordering states that every nonempty set of positive integers contains a least element; that is, there is some in such that for all in In such a case, we say that the set is well-ordered with respect to The following proof is a typical “proof by contradiction” involving the Well-Ordering Principle.

Proposition (Archimedean Property) If and are positive integers, then there exists a positive integer such that

Proof. Assume for a contradiction that and do not satisfy the statement; that is, assume there exists positive integers and such that for every positive integer Consider the set,

which consists only of positive integers. By the Well-Ordering Principle, possess a least element, say for some positive integer However, is in and is less than by

Therefore, for positive integers and there must exist a positive integer such that

    The method of proof by using the Principle Mathematical Induction is frequently useful in the theory of numbers. Familiarity with this type of argument is essential to subsequent work.

Proposition (Principle of Mathematical Induction) Let be a set of positive integers with the following property:

(i) The integer 1 belongs to

(ii) If is in then is in

Then is the set of positive integers.

Proof. Let be the set of all positive integers not in and assume that is not empty. The Well-Ordering Principle states that must have a least element, say However, by (i) 1 is in and so it follows that and and therefore, is not in Then by (ii), is in which contradicts that lies in Therefore, is empty and so is the set of positive integers.

    In any proof by mathematical induction, we must not forget to show that is in (called the basis step). Even if we show that the truth of in implies that is in if is not in then we cannot conclude that is the set of positive integers. For example, let be the set of all positive integers that satisfies: Suppose satisfies, Using this we have, so also satisfies: So, if satisfies: then, it would follow that is true for all positive integers However, since does not satisfy it is not true. In fact, is false for all positive integers
The next result states that the Well-Ordering Principle and the Principle of Mathematical Induction are equivalent. So in fact, either one can be used as an axiom.

Proposition (Mathematical Induction – Well Ordering Equivalence) If a set of positive integers has the two properties (i) the integer 1 belongs to and (ii) whenever the integer is in the next integer must also be in   then is the set of positive integers; then every set of positive integers must a have a least element.

Proof. Assume that there exists a nonempty set of positive integers, say that does not have a least element. Because is the smallest positive integer, is not in and so is smaller than all integers in Let be the set of all positive integers that are less than all the integers in At least is in and so is nonempty. Suppose that is in If is in then by the Principle of Mathematical Induction must be the set of all positive integers, and thus is empty and so every set of positive integers must have a least element. If is not in then is the least integer in which contradicts the definition of Therefore, if is a nonempty set of positive integers, then must have a least integer.

The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids (Steinhaus 1999, pp. 252-256): the cube, dodecahedron, icosahedron, octahedron, and tetrahedron, as was proved by Euclid in the last proposition of the Elements. The Platonic solids are sometimes also called “cosmic figures” (Cromwell 1997), although this term is sometimes used to refer collectively to both the Platonic solids and Kepler-Poinsot solids (Coxeter 1973).

The Platonic solids were known to the ancient Greeks, and were described by Plato in his Timaeus ca. 350 BC. In this work, Plato equated the tetrahedron with the “element” fire, the cube with earth, the icosahedron with water, the octahedron with air, and the dodecahedron with the stuff of which the constellations and heavens were made (Cromwell 1997). Predating Plato, the neolithic people of Scotland developed the five solids a thousand years earlier. The stone models are kept in the Ashmolean Museum in Oxford (Atiyah and Sutcliffe 2003).

Schläfli (1852) proved that there are exactly six regular bodies with Platonic properties (i.e., regular polytopes) in four dimensions, three in five dimensions, and three in all higher dimensions. However, his work (which contained no illustrations) remained practically unknown until it was partially published in English by Cayley (Schläfli 1858, 1860). Other mathematicians such as Stringham subsequently discovered similar results independently in 1880 and Schläfli’s work was published posthumously in its entirety in 1901 (Hovinga).

If is a polyhedron with congruent (convex) regular polygonal faces, then Cromwell (1997, pp. 77-78) shows that the following statements are equivalent.

1. The vertices of all lie on a sphere.

2. All the dihedral angles are equal.

3. All the vertex figures are regular polygons.

4. All the solid angles are equivalent.

5. All the vertices are surrounded by the same number of faces.

Let (sometimes denoted ) be the number of polyhedron vertices, (or ) the number of graph edges, and (or ) the number of faces. The following table gives the Schläfli symbol, Wythoff symbol, and C&R symbol, the number of vertices , edges , and faces , and the point groups for the Platonic solids (Wenninger 1989). The ordered number of faces for the Platonic solids are 4, 6, 8, 12, 20 (Sloane’s A053016; in the order tetrahedron, cube, octahedron, dodecahedron, icosahedron), which is also the ordered number of vertices (in the order tetrahedron, octahedron, cube, icosahedron, dodecahedron). The ordered number of edges are 6, 12, 12, 30, 30 (Sloane’s A063722; in the order tetrahedron, octahedron = cube, dodecahedron = icosahedron).

Solid	Schläfli symbol	Wythoff symbol	C&R Symbol				Group
cube		3 2 4		8	12	6
dodecahedron		3 2 5		20	30	12
icosahedron		5 2 3		12	30	20
octahedron		4 2 3		6	12	8
tetrahedron		3 2 3		4	6	4

The duals of Platonic solids are other Platonic solids and, in fact, the dual of the tetrahedron is another tetrahedron. Let be the inradius of the dual polyhedron (corresponding to the insphere, which touches the faces of the dual solid), be the midradius of both the polyhedron and its dual (corresponding to the midsphere, which touches the edges of both the polyhedron and its duals), the circumradius (corresponding to the circumsphere of the solid which touches the vertices of the solid) of the Platonic solid, and the edge length of the solid. Since the circumsphere and insphere are dual to each other, they obey the relationship

(1)

(Cundy and Rollett 1989, Table II following p. 144). In addition,

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

The following two tables give the analytic and numerical values of these distances for Platonic solids with unit side length.

solid
cube
dodecahedron
icosahedron
octahedron
tetrahedron

solid
cube	0.5	0.70711	0.86603
dodecahedron	1.11352	1.30902	1.40126
icosahedron	0.75576	0.80902	0.95106
octahedron	0.40825	0.5	0.70711
tetrahedron	0.20412	0.35355	0.61237

Finally, let be the area of a single face, be the volume of the solid, and the polyhedron edges be of unit length on a side. The following table summarizes these quantities for the Platonic solids.

solid
cube	1	1
dodecahedron
icosahedron
octahedron
tetrahedron

The following table gives the dihedral angles and angles subtended by an edge from the center for the Platonic solids (Cundy and Rollett 1997, Table II following p. 144).

solid	(rad)	()		()
cube		90.000		70.529
dodecahedron		116.565		41.810
icosahedron		138.190		63.435
octahedron		109.471		90.000
tetrahedron		70.529		109.471

The number of polyhedron edges meeting at a polyhedron vertex is . The Schläfli symbol can be used to specify a Platonic solid. For the solid whose faces are -gons (denoted ), with touching at each polyhedron vertex, the symbol is . Given and , the number of polyhedron vertices, polyhedron edges, and faces are given by

			(8)
			(9)
			(10)

The plots above show scaled duals of the Platonic solid embedded in a cumulated form of the original solid, where the scaling is chosen so that the dual edges lie at the incenters of the original faces (Wenninger 1983, pp. 8-9).

Since the Platonic solids are convex, the convex hull of each Platonic solid is the solid itself. Minimal surfaces for Platonic solid frames are illustrated in Isenberg (1992, pp. 82-83).

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