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Set Theory. • Sets Have Elements, Written X ∈ X, and Subsets, Written a ⊆ X. • the Empty Set ∅ Has No Elements

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Set Theory. • Sets Have Elements, Written X ∈ X, and Subsets, Written a ⊆ X. • the Empty Set ∅ Has No Elements Set theory. • Sets have elements, written x 2 X, and subsets, written A ⊆ X. • The empty set ? has no elements. • A function f : X ! Y takes an element x 2 X and returns an element f(x) 2 Y . The set X is its domain, and Y is its codomain. Every set X has an identity function idX defined by idX (x) = x. • The composite of functions f : X ! Y and g : Y ! Z is the function g ◦ f defined by (g ◦ f)(x) = g(f(x)). • The function f : X ! Y is a injective or an injection if, for every y 2 Y , there is at most one x 2 X such that f(x) = y (resp. surjective, surjection, at least; bijective, bijection, exactly). In other words, f is a bijection if and only if it is both injective and surjective. Being bijective is equivalent to the existence of an inverse function f −1 such −1 −1 that f ◦ f = idY and f ◦ f = idX . • An equivalence relation on a set X is a relation ∼ such that { (reflexivity) for every x 2 X, x ∼ x; { (symmetry) for every x; y 2 X, if x ∼ y, then y ∼ x; and { (transitivity) for every x; y; z 2 X, if x ∼ y and y ∼ z, then x ∼ z. The equivalence class of x 2 X under the equivalence relation ∼ is [x] = fy 2 X : x ∼ yg: The distinct equivalence classes form a partition of X, i.e., every element of X is con- tained in a unique equivalence class. Incidence geometry. • An incidence geometry consists of a set P of points and a set L of subsets of P , called lines such that { (I1) every pair of distinct points is contained in a unique line, { (I2) every line contains at least two distinct points, and { (I3) there exist three points not contained in any one line. • Two lines `1 and `2 are parallel if either `1 = `2 or `1 and `2 have no point in common. • An incidence gometry (P; L) satisfies { the elliptic parallel postulate if, for every line ` and point p not contained in `, there are no lines containing p parallel to `, { the Euclidean parallel postulate if, for every line ` and point p not contained in `, there is a unique line containing p parallel to `, and { the hyperbolic parallel postulate if, for every line ` and point p not contained in `, there are at least two lines containing p parallel to `. • Let (P1;L1) and (P2;L2) be incidence geometries. An isomorphism from (P1;L1) to (P2;L2) is a function f : P1 ! P2 such that { f is a bijection, and 1 2 { a subset A ⊆ P1 is an element of L1 if and only if the subset f(A) = ff(p): p 2 Ag is an element of L2. Affine geometry. • An affine plane is an incidence geometry satisfying the Euclidean parallel postulate. • The equivalence class of the line ` under the equivalence relation of parallelism is called the pencil of lines parallel to `. • The common number of points per line in a finite affine plane is called order of the affine plane. Projective geometry. • A projective plane is an incidence geometry in which every line contains at least three points and the elliptic parallel postulate holds, i.e., every pair of lines has a point in common. • The completion of an affine plane (P; L) is the projective plane (P; L) defined as follows. { An element of P is either an element of P or a pencil of lines in (P; L). { A subset of P is an element of L if it is either of the form `1 = f[`]: ` 2 Lg or of the form ` = ` [ f[`]g for some ` 2 L. The line `1 is called the line at infinity. The points corresponding to pencils are called ideal points. 2 • The real projective plane is the projective plane RP defined as follows. 2 { A point in RP is a line through the origin in R3. 2 { A set of lines through the origin in R3 is a line in RP if and only if the lines form a plane in R3. 2 • We also talk about RP in terms of homogeneous coordinates, in which a point is an equivalence class [x; y; z] of nonzero vector under the equivalence relation of scalar mul- tiplication, and a line is a solution set of an equation of the form ax + by + cz = 0 where a; b; c 2 R are not all zero. • The pencil of lines through a point p in a projective plane is the set of all lines containing p. • The dual of the projective plane (P; L) is the projective plane (P ∗;L∗) with P ∗ = L and L∗ the set of pencils of lines through points p 2 P . • A triangle in a projective plane is a set of three non-collinear points, called vertices. The sides of the triangle are the lines through the pairs of vertices. Two triangles are in point perspective if the three lines through corresponding pairs of vertices intersect at a common point, and they are in line perspective if the three pairs of corresponding sides intersect on a common line. 3 • Desargues' theorem, also known as the axiom P5, is the statement that, if two triangles are in point perspective, then they are in line perspective. • Pappus' theorem, also known as the axiom P6, is the statement that, given two collinear triples of points, all distinct, the lines through any three pairs of corresponding points intersect on a common line. • A projective 3-space is a triple (P; L; A) of a set P of points and sets L and A of subsets of P , whose elements are called lines and planes, respectively, subject to the following axioms. { (S1) Every two disinct points are contained in a unique line. { (S2) Three non-collinear points are contained in a unique plane. { (S3) Every line and every plane have at least a point in common. { (S4) The intersection of every pair of planes contains a line. { (S5) There are four non-coplanar points, no three of which are collinear. { (S6) Every line contains at least three points. • The projection from the point p0 to the plane X in a projective 3-space (P; L; A) is the function sending p 2 P n fp0g to the unique point of intersection of `(p0; p) with X. • Let ` and `0 be lines in a projective plane. A perspectivity from ` to `0 is any function given by projection from a point on neither to `0.A projectivity from ` to `0 is a compo- sition of perspectivities. The set of projectivities from ` to itself forms a group denoted PJ(`). • Fix a line `1 2 L.A dilatation is an automorphism of (P; L) fixing `1 pointwise. A translation is a dilatation that fixes no other points (or else the identity). A parallelogram 0 0 0 0 0 0 is a tuple (p; p ; q; q ) of points not in `1 such that `(p; p ) \ `(q; q ) and `(p; q) \ `(p ; q ) are contained in `1. Algebra. • A division ring or skew field is a set D equipped with addition and multiplication oper- ations (a; b) 7! a + b (a; b) 7! ab; subject to the following axioms. { Addition and multiplication are both associative. { Addition is commutative. { There is an additive unit 0 and a multiplicative unit 1, which are distinct. { Additive inverses exist. { Left and right multiplicative inverses of nonzero elements exist. { Multiplication distributes over addition on the right and the left. A field is a division ring with commutative multiplication. Fields are generically denoted by F. 4 • Integers a and b are congruent modulo n, written a ≡ b mod n, if a − b is divisible by n. The set of equivalence classes modulo n is denoted Z=nZ. • The affine plane over the division ring D has points pairs (x; y) of elements of D and lines solution sets to equations ax + by + c with coefficients in D, where a and b are not both 2 zero. The projective plane over D, written DP , has points equivalence classes [x; y; z] of nonzero vectors with entries in D under scalar multiplication on the right, and the lines are solution sets to nonzero equations of the form ax + by + cz = 0 with coefficients in D. • An element d 2 D is central if dd0 = d0d for every d0 2 D. • The projective general linear group over D, denoted P GLn(D), is the set of equivalence classes of n × n invertible matrices with coefficients in D up to scalar multiplication by central elements of D. • A group is a set G equipped with a multiplication operation that is associative and has a unit and inverses for all elements. The order of the group is the number of elements in G. • Let F be a field. A linear fractional transformation over F is a function of the form at+b f(t) = ct+d , where a; b; c; d 2 F and ad − bc 6= 0..
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