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From Wikipedia

Cartesian product

In mathematics, a Cartesian product (or product set) is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept.

Specifically, the Cartesian product of two sets X (for example the points on an x-axis) and Y (for example the points on a y-axis), denoted X× Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y (e.g., the whole of the x–y plane):

X\times Y = \{(x,y) | x\in X \ \text{and} \ y\in Y\}.

For example, the Cartesian product of the 13-element set of standard playing card ranks {Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2} and the four-element set of card suits {♠, ♥, ♦, ♣} is the 52-element set of all possible playing cards: ranks× suits = {(Ace, ♠), (King, ♠), ..., (2, ♠), (Ace, ♥), ..., (3, ♣), (2, ♣)}. The corresponding Cartesian product has 52 = 13 × 4 elements. The Cartesian product of the suits× ranks would still be the 52 pairings, but in the opposite order {(♠, Ace), (♠, King), ...}. Ordered pairs (a kind of tuple) have order, but sets are unordered. The order in which the elements of a set are listed is irrelevant; you can shuffle the deck and it's still the same set of cards.

A Cartesian product of two finite sets can be represented by a table, with one set as the rows and the other as the columns, and forming the ordered pairs, the cells of the table, by choosing the element of the set from the row and the column.

Basic properties

Let A, B, C, and D be sets.

In cases where the two input sets are not the same, the Cartesian product is not commutative because the ordered pairs are reversed.

Although the elements of each of the ordered pairs in the sets will be the same, the pairing will differ.

A \times B \neq B \times A

For example:

{1,2} x {3,4} = {(1,3), (1,4), (2,3), (2,4)}

{3,4} x {1,2} = {(3,1), (3,2), (4,1), (4,2)}

One exception is with the empty set, which acts as a "zero", and for equal sets.

A \times \emptyset = \emptyset \times A = \emptyset

and, supposing G,T are sets and G=T:

(G) \times (T) = (T) \times (G).

Strictly speaking, the Cartesian Product is not associative.

(A\times B)\times C \neq A \times (B \times C)

The Cartesian Product acts nicely with respect to intersections.

(A \cap B) \times (C \cap D) = (A \times C) \cap (B \times D)

Notice that in most cases the above statement is not true if we replace intersection with union.

(A \cup B) \times (C \cup D) \neq (A \times C) \cup (B \times D)

However, for intersection and union it holds for:

(A) \times (B \cap C) = (A \times B) \cap (A \times C)

and,

(A) \times (B \cup C) = (A \times B) \cup (A \times C).

n-ary product

The Cartesian product can be generalized to the n-ary Cartesian product over n sets X1, ..., Xn:

X_1\times\cdots\times X_n = \{(x_1, \ldots, x_n) : x_i \in X_i \}.

It is a set of n-tuples. If tuples are defined as nested ordered pairs, it can be identified to (X1× ... × Xn-1) × Xn.

Cartesian square and Cartesian power

The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product X2 = X× X. An example is the 2-dimensional planeR2= R × R where R is the set ofreal numbers - all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).

The cartesian power of a setX can be defined as:

X^n = \underbrace{ X \times X \times \cdots \times X }_{n}= \{ (x_1,\ldots,x_n) \ | \ x_i \in X \ \text{for all} \ 1 \le i \le n \}.

An example of this is R3= R × R × R, with R again the set of real numbers, and more generally Rn.

The n-ary cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.

See also:

Infinite products

It is possible to define the Cartesian product of an arbitrary (possibly infinite) family of sets. If I is any index set, and {Xi | i∈ I} is a collection of sets indexed by I, then the Cartesian product of the sets in X is defined to be

\prod_{i \in I} X_i = \{ f : I \to \bigcup_{i \in I} X_i\ |\ (\forall i)(f(i) \in X_i)\},

that is, the set of all functions defined on the index set such that the value of the function at a particular index i is an element of Xi .

For each j in I, the function

\pi_{j} : \prod_{i \in I} X_i \to X_{j},

defined by Ï€j(f) = f(j) is called the j -th projection map.

An important case is when the index set is N the natural numbers: this Cartesian product is the set of all infinite sequences with the i -th term in its corresponding set Xi . For example, each element of

\prod_{n = 1}^\infty \mathbb R =\mathbb{R}^\omega= \mathbb R \times \mathbb R \times \cdots,

can be visualized a

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.

The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768-1822), although they were first described by Norwegian-Danish land surveyor and mathematician Caspar Wessel (1745-1818). Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane.

The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates– the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation.

Notational conventions

In complex analysis the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts, like this:

z = x + iy\, for example: z = 4 + i5,

where x and y are real numbers, and i is the imaginary unit. In this customary notation the complex number z corresponds to the point (x, y) in the Cartesian plane.

In the Cartesian plane the point (x, y) can also be represented in polar coordinates as

(x, y) = (r\cos\theta, r\sin\theta)\qquad\left(r = \sqrt{x^2+y^2}; \quad \theta=\arctan\frac{y}{x}\right).\,

In the Cartesian plane it may be assumed that the arctangent takes values from −Ï€/2 to Ï€/2 (in radians), and some care must be taken to define the real arctangent function for points (x, y) when x≤ 0. In the complex plane these polar coordinates take the form

z = x + iy = |z|\left(\cos\theta + i\sin\theta\right) = |z|e^{i\theta}\,

where

|z| = \sqrt{x^2+y^2}; \quad \theta = \arg(z) = -i\ln\frac{z}.\,

Here |z| is the absolute value or modulus of the complex number z; θ, the argument of z, is usually taken on the interval 0 ≤ θ< 2π; and the last equality (to |z|eiθ) is taken from Euler's formula. Notice that the argument of z is multi-valued, because the complex exponential function is periodic, with period 2πi. Thus, if θ is one value of arg(z), the other values are given by arg(z) = θ + 2nπ, where n is any integer ≠ 0.

The theory of contour integration comprises a major part of complex analysis. In this context the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the positive direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point z = 1, then travel up and to the left through the point z = i, then down and to the left through −1, then down and to the right through −i, and finally up and to the right to z = 1, where we started.

Almost all of complex analysis is concerned with complex functions– that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here it is customary to speak of the domain of f(z) as lying in the z-plane, while referring to the range or image of f(z) as a set of points in the w-plane. In symbols we write

z = x + iy;\qquad f(z) = w = u + iv\,

and often think of the function f as a transformation of the z-plane (with coordinates (x, y)) into the w-plane (with coordinates (u, v)).

Stereographic projections

It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place it's center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole is "above" the plane.

We can establish a one-to-one correspondence between the points on the surface of the sphere minus the north pole and the points in the complex plane as follows. Given a point in the plane, draw a straight line connecting it with the north pole on the sphere. That line will intersect the surface of the sphere in exactly one other point. The point z = 0 will be projected onto the south pole of the sphere. Since the interior of the unit circle lies inside the sphere, that entire region (|z| < 1) will be mapped onto the southern hemisphere. The unit circle itself (|z| = 1) will be mapped

Horizontal plane

In geometry, physics, astronomy, geography, and related sciences and contexts, a planeis said to be horizontal at a given point if it is locally perpendicular to thegradient of the gravityfield, i.e., with the direction of the gravitational force (per unit mass) at that point.

In radio science, horizontal plane is used to plot an antenna's relative field strength in relation to the ground (which directly affects a station's coverage area) on a polar graph. Normally the maximum of 1.000 or 0 dB is at the top, which is labeled 0o, running clockwise back around to the top at 360°. Other field strengths are expressed as a decimal less than 1.000, a percentage less than 100%, or decibels less than 0 dB. If the graph is of an actual or proposed installation, rotation is applied so that the top is 0otrue north. See also the perpendicular vertical plane.

In general, something that is horizontal can be drawn from left to right (or right to left), such as the x-axis in the Cartesian coordinate system.

Discussion

Although the word horizontal is common in daily life and language (see below), it is subject to many misconceptions. The precise definition above and the following discussion points will hopefully clarify these issues.

  • The concept of horizontality only makes sense in the context of a clearly measurable gravity field, i.e., in the 'neighborhood' of a planet, star, etc. When the gravity field becomes very weak (the masses are too small or too distant from the point of interest), the notion of being horizontal loses its meaning.
  • In the presence of a simple, time-invariant, rotationally symmetric gravity field, a plane is horizontal only at the reference point. The horizontal planes with respect to two separate points are not parallel, they intersect.
  • In general, a horizontal plane will only be perpendicular to a vertical direction if both are specifically defined with respect to the same point: a direction is only vertical at the point of reference. Thus both horizontality and verticality are strictly speaking local concepts, and it is always necessary to state to which location the direction or the plane refers to. Note that (1) the same restriction applies to the straight lines contained within the plane: they are horizontal only at the point of reference, and (2) those straight lines contained in the plane but not passing by the reference point are not horizontal anywhere.
  • In reality, the gravity field of a heterogeneous planet such as Earth is deformed due to the inhomogeneous spatial distribution of materials with different densities. Actual horizontal planes are thus not even parallel even if their reference points are along the same vertical direction.
  • At any given location, the total gravitational force is a function of time, because the objects that generate the reference gravity field move relative to each other. For instance, on Earth, the local horizontal plane at a given point (as materialized by a pair of spirit levels) changes with the relative position of the Moon (air, sea and land tides).
  • Furthermore, on a rotating planet such as Earth, there is a difference between the strictly gravitational pull of the planet (and possibly other celestial objects such as the Moon, the Sun, etc.), and the apparent net force applied (e.g., on a free-falling object) that can be measured in the laboratory or in the field. This difference is due to the centrifugal force associated with the planet's rotation. This is a fictitious force: it only arises when calculations or experiments are conducted in non-inertial frames of reference.

Practical use in daily life

The concept of a horizontal plane is thus anything but simple, although, in practice, most of these effects and variations are rather small: they are measurable and can be predicted with great accuracy, but they may not greatly affect our daily life.

This dichotomy between the apparent simplicity of a concept and an actual complexity of defining (and measuring) it in scientific terms arises from the fact that the typical linear scales and dimensions of relevance in daily life are 3 orders of magnitude (or more) smaller than the size of the Earth. Hence, the world appears to be flat locally, and horizontal planes in nearby locations appear to be parallel. Such statements are nevertheless approximations; whether they are acceptable in any particular context or application depends on the applicable requirements, in particular in terms of accuracy.

In graphical contexts, such as drawing and drafting on rectangular paper, it is very common to associate one of the dimensions of the paper with a horizontal, even though the entire sheet of paper is standing on a flat horizontal (or slanted) table. In this case, the horizontal direction is typically from the left side of the paper to the right side. This is purely conventional (although it is somehow 'natural' when drawing a natural scene as it is seen in reality), and may lead to misunderstandings or misconceptions, especially in an educational context.


Projective plane

In mathematics, the projective plane is a geometric construction that extends the concept of a plane. In the ordinary plane, two lines typically intersect in a single point, but there are some pairs of lines — namely, parallel lines — that do not intersect. The projective plane is, in one view, the ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two lines in the projective plane intersect in one and only one point.

The projective plane has two common definitions. The first comes from linear algebra; it produces planes that are homogeneous spaces for some of the classical groups. Important examples include the real projective plane \mathbb{RP}^2 and the complex projective plane \mathbb{CP}^2. The second, more general definition comes from axiomatic geometry and finite geometry; it is suitable for study of the incidence properties of plane geometry.

The projective plane generalizes to higher-dimensional projective spaces; that is, a projective plane is a 2-dimensional projective space.

Linear-algebraic definition

In one view, the projective plane is the set of lines through the origin in 3-dimensional space, and a line in the projective plane arises from a plane through the origin in 3-dimensional space. This idea can be made precise as follows.

Let K be any division ring. Let K3 denote the set of all triples x = (x0, x1, x2) of elements of K (a Cartesian product). For any nonzero x in K3, the line in K3 through the origin and x is the subset

\{k x : k \in K\}

of K3. Similarly, let x and y be linearly independent elements of K3, meaning that if k x + l y = 0 then k = l = 0. The plane through the origin, x, and y in K3 is the subset

\{k x + l y : k, l \in K\}

of K3. The plane contains various lines.

The projective plane over K, denoted K\mathbb{P}^2, is the set of all lines in K3. A subset L of K\mathbb{P}^2 is a line in K\mathbb{P}^2 if there exists a plane in K3 whose set of lines is exactly L.

A slightly different definition is as follows. The projective plane is the set K3 - {(0, 0, 0)} modulo the equivalence relation

x \sim k x, k \in K.

Lines in the projective plane are defined exactly as above. If K is a topological space, then K\mathbb{P}^2 inherits a topology via the product, subspace, and quotient topologies.

The coordinates (x0, x1, x2) on K\mathbb{P}^2 are called homogeneous coordinates. Each triple (x0, x1, x2) represents a well-defined point in K\mathbb{P}^2, except for the triple (0, 0, 0), which represents no point. Each point in K\mathbb{P}^2 is potentially represented by many triples.

Examples

The real projective plane \mathbb{RP}^2 arises when K is taken to be the real numbers. As a closed, non-orientable real 2-manifold, it serves as a fundamental example in topology.

The complex projective plane \mathbb{CP}^2 arises when K is taken to be the complex numbers. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other fields serve as fundamental examples in algebraic geometry.

The quaternionic projective plane is also of independent interest. The Cayley plane is considered to be a projective plane over the octonions, but the preceding construction does not suffice to describe it, because the octonions do not form a division ring.

Taking K to be the finite field of pn elements produces a projective plane of p2 n + pn + 1 points. The Fano plane, discussed below, is the example with pn = 2.

Relationship to the ordinary plane

The ordinary plane K2 over K embeds into K\mathbb{P}^2 via the map

(x_1, x_2) \mapsto (1, x_1, x_2).

The complement of the image is the set of points of the form (0, x1, x2). From the point of view of the embedding just given, these points are points at infinity. They constitute a line in K\mathbb{P}^2 — namely, the line arising from the plane

\{k (0, 0, 1) + l (0, 1, 0) : k, l \in K\}

in K3. Intuitively, the points at infinity are the "extra" points where parallel lines intersect; the point (0, x1, x2) is where all lines of slope x2 / x1 intersect. Consider for example the two lines

a = \{(x_1, 0) : x_1 \in K\},
b = \{(x_1, 1) : x_1 \in K\}

in the ordinary plane K2. These lines have slope 0 and do not intersect. They can be regarded as subsets of K\mathbb{P}^2 via the embedding above, but these subsets are not lines in K\mathbb{P}^2. Add the point (0, 1, 0) to each subset; that is, let

\bar a = \{(1, x_1, 0) : x_1 \in K\} \cup \{(0, 1, 0)\},
\bar b = \{


From Yahoo Answers

Question:To me questions like these force me to think outside the box. Our history and our universe is on one timeline. We wake up, live our day, then go to sleep. This is what we know and we discover all of what we know in this timeline about the world surrounding us and inside of us. I consider this timeline to be viewed in one dimension. Has anyone considered the possibility of more than one timeline on seperate axis' existing all around our timeline? i.e. Parallel dimensions. What is the best theories on parallel dimensions? Are gaps between timelines considered? Inter-timeline travel? -Properties concerning each timeline - A timeline (us) One time interval, Days are 24-hours - B timeline, objects could dissapear and reappear in one time interval, while living organisms exist in another time interval. -C timeline The universe is growing and collapsing at the same time.

Answers:Time is one dimension, one of the famous four: height, width, depth, time, using commonplace terms. I could also write (i, j, k, t); where i, j, and k are unit vectors (e.g., i dot i = 1) designating the three spatial dimensions. And t would be the fourth dimension. Time is real...it's not just the passage of events, like rising, showering, breakfasting, etc. Time passes even if no events take place. Time can be stretched out so that, for example, it would take 2 Earth seconds for 1 second to tick off on a very fast spaceship. Such a stretch is called dilation and this phenomenon demonstrates that time can be manipulated. That is a prime clue that time is real. If it were not real, we wouldn't be able to dilate it. This dilation can be used to travel into the future. For instance, if a star trekker in the above example traveled one year according to his clock on the spaceship and then returned to Earth, he would find Earth time had advanced two years. In other words, the spaceman would find himself one year into his future when he stepped out of the ship. As to "parallel dimensions" you are mixing concepts. In fact it's higher dimensions and parallel universes. [See source.] String theory posits up to 11 dimensions instead of the conventional four we know and love. One aspect of the theory suggests the other seven (all spatial) are simply curled up so tiny (1 Planck length = 10^-33 cm) that we can't see them. But strings, because they are also tiny, can see the extra dimensions and are constrained by them just as we are constrained by the four dimensions of our universe. One WAG of string theory is the parallel universe. Each universe is like a slice of bread in a mega universe loaf. Each slice is separated by 1 Planck length and 1 Planck time (which is also very tiny but I've forgotten the number). One SWAG resulting from the WAG is that two or more of the parallel universes collided and rebound. That change in momentum over time gave rise to the tremendous energy we call the Big Bang. Thus, there would be a BB in our universe as well as another BB in the universe that collided with us. And so, if we count the BB as t = 0, the timeline of the two BBs starts at the same time the two parallel universes. That is not to say the chains of events are identical...it's unlikely they are. But time, a real dimension, will be identical. There would be a gap between the timelines of the two parallel universes. The time length of that discontinuity would be 1 Planck time. And there would be a spatial gap of 1 Planck length between the two after they rebound from the collision. Both these gaps result because, theorectically, the 1 Planck time and the 1 Planck length are the smallest possible intervals in time and space. Unless the makeup of the two colliding universes was significantly different, the makeup of the two BBs ought to be about the same. So the uniform initial energies of both would go through the same evolutions and end up with the same kind of galaxies, planets, and energies. This suggests there might be living, intelligent beings living out their lives in the parallel universe...wondering if there is life out there.

Question:the cartesian plane, specifies that both the x-coordinate and the y-coordinate in quadrant one, are always positive. Explain and prove why the points in the other thee quadrants have a different outcome?

Answers:The points in the other three quadrants have a different outcome, because in all of these, at least one co-ordinate is negative. In quadrant II, x is negative, y is positive. In quadrant iii, both co-ordinates are negative. In the fourth quadrant, x is positive, y is negative. NB: For a better answer, ask in the math section.

Question:Can someone show me how I would do this? It asks me to calculate the distance between the points F = (2,2) and M = (5,-4) in the cartesian plane. And to write the answer in radical form. I'm just not sure how I would go about doing this. Could someone please help me out? 10 points will be awarded to the best answer!

Answers:it's a lot like computing slope in that you find the differences between the x's and between the y's, but then you square them, add them, and square root the result: d = [ (x1 - x2) + (y1 - y2) ] d = [ (5-2) + (-4-2) ] d = ( 9 + 36) d = 45 = 3 5

Question:read title i need it today

Answers:Hilarious... I think. I know you needed this yesterday, or the day before, but here's an idea or two. A parody of the X-Men - the "X" is really a slanted Cartesian plane anyways, isn't it? Passion of the Cartesian Plane - Jesus being crucified on the C-plane is already causing controversy, push the envelope a bit farther. C-Plane Soaps - create torrid love triangles (Oblique, Equilateral, Isosceles, and Scalene) where drama erupts between different points. That's all I got for now - hope I helped a bit! :)

From Youtube

The Equation of a Plane Part 1.avi :This lesson develops the vector, parametric and scalar (or Cartesian) equations of planes in Three - Space. This video covers sections 8.2 and 8.3 in the Mcgraw Hill MCV4U text. The lesson also includes some examples demonstrating how to use or create these equations given points and vectors in the plane or a point in the plane and a normal vector or 3 points in the plane. This is the first part of a three part lesson. This lesson was created for the Calculus and Vectors (MCV4U) course in the province of Ontario, Canada.

The Language of Mathematics (23): Cartesian Coordinate System :Table of contents listing all instructional math videos available from Chycho TV at www.chycho.com List of videos that are available for download at www.chycho.com INTRODUCTION: www.chycho.com During the last few years the education systems in Canada and the United States has been eroding. Students have been crammed into larger class sizes while the quality of teaching has deteriorated, in large part due to a lack of funding from local and federal governments. John Ralston Saul, one of Canada's most celebrated authors and essayist, has warned us that "Canada's democracy... is eroding with every dime deducted from education budgets." I have stated previously that I believe "as a collective we must understand that democracy can only exist in a society with an educated populace, and the right for self-governance can only be obtained through knowledge." I also believe that as long as the Internet remains Neutral, we, as a global community, can make a difference. We can fill the gap left behind by our governments and be proactive in educating our children Sharing our communal experiences and knowledge is the essence of the Internet and chycho.com is proud to be a part of this movement. The information contained and conveyed through this medium is what will shape our future, and this site was created to represent a unique personal perspective and to contribute in bringing about positive changes. As a result I have decide to produce instructional math videos to help those that ...