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Free Forum # 3 multiset

In mathematics a multiset (also called a pocket or bag, in the original) differs from a set in which each member of it has an associated multiplicity (a natural number) indicating how often the item is a member of the group, for example, in the multiset {a, a, b, b, b, c}, the multiplicities members a, b and c are 2, 3 and 1, respectively. Formal definition

In set theory, a multiset is defined as the pair (A, m) where A is a set and m: A → N is an application of A to N (positive integers). A is called the underlying set of elements. For each to A, the multiplicity of a is the number m (a).
is common to write the function m as a set of ordered pairs {(a, m (a)): a ∈ A}. Since this is, without doubt, the definition (using set theory) of the function m. For example:
• The multiset written as {a, b, b} is defined as {(a, 1), (b, 2)}
• The next {a, a, b}, in turn, is defined as {(a, 2), (b, 1)}, and
• Finally, the multiset {a, b} is defined as {( a, 1), (b, 1)}.
If the set A is finite, the size or length of the multiset (A, m) is the sum of all multiplicities for each element of A:

A submulticonjunto (B, n) of the multiset (A, m) is a subset B ⊆ A and an application n: B → N such that n (a) ≤ m (a).


1. Specific example of how it is applied on a multiset data structure
2. What language do you think you can give a multiset?

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"descriptive geometry" THEORY CLASS TUESDAY 30 - JUNE - 2009

So far we have studied the known Plane Geometry. Then study what is the geometry, which is primarily intended to teach and must be represented in a flat projection on a three-dimensional object that can be derived from its shape, dimensions and elements that constitute it.

Descriptive Geometry is the science that aims to study the extent considered under its three dimensions: line, surface and volume. Descriptive Geometry is for the drawing that the grammar for a language.

PROJECTION: is a graphical representation, by whatever method of representing an object, such as we see it. The points of intersection between the plane and rays projection from the point of observation and go through the different points of view, are the projection of the object.

In any projection involves three elements:

- The point of observation, eye, light source or point of view. From the vantage point of projection rays are emitted, which passes through all the points of the object, penetrating the surface projection (projection of the object).

- The object being observed.

- The surface of projection screen, flat box, etc.

A. TYPES OF PROYCCIÓN.

By varying the point of observation:

1 .- Conic projection: when the observation point is in the finite (or relatively close to the object projected ), all projection rays start from the same point observation and form a kind of cone.

2 .- Cylindrical Projection: is one in which the viewpoint is at infinity, all projection rays are parallel to each other and the cone obtained in the previous case
becomes a cylinder.

varying the projection surface:

1 .- Projection

Oblique projection when the rays of an angle either, other than 90 degrees with the plane of projection.

2 .- Orthogonal Projection: projection when the rays are at right angles (90 °) to the projection plane.

3 .- Projected Outlook: Every projection in one view represents a body in three dimensions, with the way ydisposición as we see in space.

B. Projection plane.

planes are undefined, but may be limited by their traces or lines of intersection with each other. The line of intersection of a vertical plane with a horizontal line called Earth.

The plans are named with Greek letters a, b, c, etc.

If two planes perpendicular to each cut are called dihedral angle, if there are three perpendicular planes which intersect or coincide at the same point, forming the trihedral angle. If more than three planes which coincide at one point, form the polyhedral angle.

projection planes can be infinite, but there are three major.

- Horizontal Plane: plant also called object or floor plan.

- Plano Vertical: also called standard.

- Plano Side-view or profile, which can be left or right.

land lines or traces of the three planes are the three Cartesian axes X, Y, Z.

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CONIC SECTIONS PRACTICE, TUESDAY CLASS OF 23 - JUNE - 2009

PRACTICE TO BE DELIVERED THE DAY TUESDAY, JUNE 30, NO FOUL.

1 .- Build an ellipse given its two axes. AB is horizontal and CD is 9 and 5.
Scale = 1 / 100

2 .- Build an ellipse given the major axis is horizontal, the two foci F and F 'and the points P1, P2, P3, P4, P5 and P6.
A = 1.5, 0
B = 8.5, 0
F = 2.5, 0
F '= 7.5, 0
P1 = 5.4, 0
P2 = 5.8, 0
P3 = 6.3, 0
P4 = 6.5, 0
P5 = 6.8, 0
P6 = 7.2, 0
SCALE = 1 / 75

3 .- Building a guideline parabola given the vertex and axis EE 'is horizontal.
A = 0.3, 0
B = 0.3, 1.5
C = 0.3, -1.5
E = 0.8, 0
E '= 2.3, 0
SCALE = 1 / 25

4 .- Build a hyperbola given the AA axis 'and the asymptotes BB' and CC 'forming an angle of 60 ° each with the axis AA'.
A = 5, 0
A '= 11, 0
SCALE = 1 / 125

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"conic sections" TUESDAY CLASS OF 23 - JUNE - 2009

METHODS OF CONSTRUCTION OF GEOMETRIC CURVE:

1 .- Method to draw an ellipse given its two axes.
- Plot the major axis AB and CD the minor axis, perpendicular to each other by the middle of both axes.
- Making center in the middle "M", we draw two concentric circles with radii equal to MA and MC.
- then divide the two circles in a pair of equal parts. In this case were 12 equal parts.
- On the big points on the circle, draw lines parallel to the axis CD.
- For the points on the circle small to draw lines parallel to the axis AB.
- Where the lines belonging to the same point in two intersecting circles is a point through which passes the curve of the ellipse.
- Repeat this with all the points of two circles.
- Finally a staff of curves joining the points of the ellipse. 2 .-

METHOD GIVEN to draw an ellipse the major axis AB & BULBS FYF '.
- Draw the major axis AB and was seeking the midpoint M and he draws a line perpendicular to AB.
- The axis AB are located at points F and F '.
- are marked several arbitrary points between O and F '.
- Making center F and F 'on OA opening and arches mark to cut the line perpendicular to AB, which results in points C and D, points of the minor axis of the ellipse.
- After making center at F and F 'on opening A1 and arches mark.
- After making center at F and F 'on and B1 mark opening arcs.
- Where they cut the arches are the first points through which the ellipse.
- Do the same with the remaining points between O and F ', to get all points of the ellipse.
- Finally a staff of curves joining the points of the ellipse. 3 .-

METHOD FOR DRAWING A PARABLE BC GUIDELINE GIVEN THE GO RTIC "A" AND AXLE EE '.
- the vertex is located, BC guideline is drawn and plotted the axis EE '.
- Making centered on "E" and EA is opening an arc to cut the shaft and locate the point "F" which is the focus of the parabola.
- are marked several arbitrary points between E and E '.
- parallel lines are drawn to the guidance for each of the arbitrary points between E and E '.
- Making center "F" and opening A1 describes arcs which cut the line that passes through the point 1 on both sides of the shaft getting the points G and G '.
- Do the same with the remaining points between E and E ', to get all points of the parabola.
- Finally a staff of curves joining the points of the parabola. 4 .-

METHOD GIVEN TO DRAW A hyperbola asymptotes BB 'and CC' AND AXLE AA '.
- Draw the axis AA 'and was looking for the midpoint M.
- By the mid-point M draw the 2 asymptotes.
- For the points A and A 'draw lines parallel to the asymptotes to cut to the asymptotes in the extension, getting the points D0, E0, F0 and G0.
- After opening MD0 is copied to this distance on each of the two asymptotes from the points D0, E0, F0 and G0.
- For each of the items found are drawn parallel to the asymptotes.
- then you'll be looking at the segment midpoint AD0 and draw a parallel to the asymptote, obtaining the point A1.
- To get the other points that make up the parable is the same on.
- then join all the dots with a curved template.

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"conic sections" THEORY CLASS TUESDAY 23 - JUNE - 2009

conic sections are called to the curved sections generated from the intersection of any plane with a cone of revolution.

Conic sections are divided into:

Ellipse: This is a closed curve, symmetric flat arising from the intersection of a cone with a plane is not parallel to the base or the generators. The sum of the distances of a any point to the foci is constant and equal to the long axis.

Parabola: a curve is flat, open and symmetrical about its axis, whose origin is the intersection of a cone by a plane parallel to one of its lines. All points of a parabola are equidistant from a fixed point "F", called the focus and a fixed line "D" called the directrix.

Hyperbola: A plane curve, open and symmetric derivative of the intersection of a cone with a plane parallel to its axis. The difference in each of their points to other fixed on its axis, called foci, is constant at a fixed distance.

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"geometric curve" TUESDAY CLASS 16 - JUNE - 2009

METHOD TO BUILD A GIVEN OVAL Axis.
Draw the axis AB.
Then the major axis is divided into four equal parts, with the method of the compass. Getting the points C, M and N. Taking the radio
MA is center M, and N to describe two circles tangent to each other. Using
center M and N, MN opening, two arcs are drawn we define the points P and Q. Plot
from P and Q lines that pass through the points M and N and extend to determining circumference cut points T, V, R and S. Using
center P and Q and QT opening and describes the arches PS and RS TV. Being defined in the oval.

METHOD TO BUILD A SMALL OVAL BECAUSE THE SHAFT. Draw the
minor axis AB.
Draw the midpoint of segment AB, the method of the compass, and continues the line. Using
center M and AM opening is plotted on the axis distance AM on both sides, getting the points C and D.
ABCD points are joined, forming a square and extend the sides of the square. Using
center A and B, AB opening describes the arc PQ and ST. Using
center C and D, with opening CP describes the PS and QT arches. Being defined in the oval.

method for constructing a football GIVEN the minor axis.
Draw the minor axis AB.
Draw the midpoint of segment AB, the method of the compass and continues the line. Using
center M and AM opening draw a circle that cuts the axis at points C and F. Joining points
AF and BF, extend both lines. Using
center A and B, AB draw two arches opening to cut the lines AF and BF, getting the points G and E
Using F center and draw an arc opening FG GE. The ball being defined.

METHOD TO BUILD A SPIRAL OF TWO CENTERS.
Given the centers A and B, joining them with a line that extends from side to side.
We plot the midpoint of segment AB, the method of the compass.
With center M and MA opening trace the arc AB.
With center opening in A and AB draw the arc BC.
with center at B and draw the arc BC opening CD.Se spiral continues alternating the center A and B, on the radius from the center to the free end of the arc.

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geometric curves, THEORY TO THE CASE OF TUESDAY, JUNE 16, 2009.

geometric curves are called curved lines can be built following a design process can be repeated exactly comfortable mind.
geometric curves are divided into:

OVALS:
is a closed smooth curve, which consists of four quarters of circle tangent to each other and symmetrical with respect to two perpendicular axes of different length.
The major axis is the longest chord that passes by the center.
The minor axis is perpendicular to the major axis through the center. Ovid

:
flat is a closed curve, which is composed of four arcs: two are equal and both are mixed.

SPIRAL:
curves are flat and open. Its generating point revolves around another point, he continually away.
The fixed point called the center.
One complete turn of the loop is called spiral. The distance between the two coils is called step closer.

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"star polygon" CLASS TUESDAY, JUNE 09, 2009

1 .- method of dividing the circle into 5 equal parts. (METHOD OF REGULAR PENTAGON).
.- Draw the circle.
.- Are drawn perpendicular diameters of the circle, 1B, Rm.
.- Find the midpoint of the radius RO, obtaining the point F. .- Using
center F and F1 radio to draw a short radius arc at the point H. Om
.- With center B and radius BH trace an arc that cuts the circle at point A.
.- The line AB is equal to one-fifth of the circle (side of a regular pentagon inscribed in circle).
.- To draw the 5-pointed star polygon, connect the dots left by 1 point, so that it begins and ends at the same point. 2 .-

method for dividing the circumference at 3, 6 OR 12 PART S SAME.
.- Draw a circle.
.- Plot perpendicular diameters of the circle, getting the points MN and OP. .- Using
center M and radius aperture equal to the circumference an arc cutting the circle at 2 points. We do the same from the point N, O and P. thus the circle is divided into 12 equal parts.
.- To draw the 12-pointed star polygon, connect the dots, leaving 4 points in between, so that it begins and ends at the same point. 3 .-

method for dividing the circle into 4, 8 or 16 equal parts.
.- Draw a circle.
.- perpendicular diameters is plotted of the circle, getting the points cough MN and OP.
.- diameters are drawn at 45 degrees of the circumference, resulting in QR and ST points.
.- 2 consecutive points are joined and are located at the midpoint of that segment. When the midpoint line cutting the circle gives the size of the 16 th of the circumference.
.- Repeat this size in the spaces between the points already exist, the circle being divided into 16 points.
.- To draw the star polygon of 16 points, connect the dots, leaving 4 points in between, so that it begins and ends at the same point.

4 .- TO DIVIDE THE CIRCLE Into 7 equal parts.
.- Draw a circle.
.- Draw the vertical diameter of the circle, getting the points MN. .- Using
center M and radius aperture equal to the circumference an arc cutting the circumference at points 1D.
.- We join the points 1 and D, where the line joining these two points crosses the line mn is the point F. Using center
.- 1 and 1F opening an arc that intersects the circle which results in point 2.
.- The opening 1-2 is the seventh of the circle, copy this distance from point 2 to get the points missing ..- To draw the 7-star polygon points, connect the dots left by 1 point, so that it begins and ends at the same point.

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"star polygon" THEORY CLASS 09-JUN-2009

The star polygon whose sides are surfaces incoming and outgoing angles alternately.

technical drawing for the star polygon can be constructed in two ways, by the method of reduction or the extension method.

Reduction Method: draw the star is inscribed within the polygon.

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Method Extension: involves using the regular polygon as the center, by plotting points of the star by extending the sides of regular polygon.
.
two cases can be found in the stellate polygone:
1, in which case the polygon is drawn through a continuum that begins and ends at the same point. This will occasionally happen when the number of vertices of the star are prime numbers.
.
Case 2 in which the polygon is drawn through two or more regular polygons inscribed and intercepted in the circle. This will occasionally happen when the number of vertices of the star have a common divisor.


Polygons CLASSIFIED BY NUMBER OF SIDES:

· Of 5 points.
· Of 10 points.
· Of 6 points.
· Of 11 points.
· From 7 points.
· Of 12 points.
· From 8 points.
· From 9 points.
• And so on.

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"BUILDING POLYGONS" CLASS OF 02-JUN-2009

1. Construct a pentagon given the side AB.

· Drawing the line AB, being equal to the side of the pentagon, extend the seg ment indefinitely from B.
· Doing center at A and B and AB are plotted opening two arcs.
· Tazar a perpendicular to segment AB by the end of B, until being cut with the arc path and get the point F.
· It determines the midpoint of AB.
· Doing center M and MF are opening an arc that intersects the extension of the segment, getting the point E.
· Doing center at A and B and AE opening two arcs are drawn, which cut the former at the points N and G.
· Doing center in N and G and AB are plotted opening two arcs, which intersect at point H. Connecting the Dots
· AN, NH, HG, GB.

2. Construct a pentagon given the circle that contains it.

· We trace the circumference, then we trace the perpendicular diameters of circumference, Bf and Rm.
· Making within R & flip RO cut the circle, to find the midpoint of the radius RO, obtaining the point F.
· Doing center F and FB radio cuts draw an arc radius at the point H. Om
• With center B and radius BH trace an arc that cuts the circle at point A.
• The line AB corresponds to the side of a regular pentagon inscribed in circumference. Connecting the Dots
· AB, BC, CD, DE and EA.

3. Construct a hexagon given the side AB.

· Drawing the line AB, being equal to the side of the hexagon.
· Doing center at A and B and AB are plotted opening two arcs whose intersection is the center of the circle inscribing the hexagon.
· Doing center O and OA opening draw the circle. • With opening
tazan AB on the circle that distance from B. · Combine all the points to close the polygon.

4. Construct a heptagon given the circumference that contains it.

. Draw the circle, then draw the vertical diameter of the circle , 1A.

· Doing center opening in A and AO, is an arc that cuts the circle, taking the points ob
d, c and n .

. Using 1 center and opening cn is equal to the side of the heptagon, divide the circumference.

. Combine all the points to close the polygon.

5. General method for construction of regular polygons from 6 to 12 sides, knowing the length of the side .

· Since the segment AB, with center at A and B and AB opening two arcs that intersect at point O.

· Doing center O and draw a semicircle aperture OA.

· draw a perpendicular through the midpoint of segment AB intersecting the semicircle,
getting the point E.

· Split the segment OE in six (6) equal parts, from E.

· Doing center opening 6 and 6-5, 6-4, 6-3, 6-2, 6-1, 6-E, we get the points 7, 8, 9, 10, 11 and 12.

• When do we draw a center Heptagon 7 and 7-A radio, whether it is an octagon in the center is 8 and Radio 8-A, if a enneagon is centered at the 9 and 9-A radio.

• The example we gave is an enneagon. Using center opening 9 and 9-A we draw a circle on which we will copy the segment AB nine consecutive times.

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"Regular Polygons" CLASS OF 02-JUNE- 2009

polygon is called the flat portion bounded by a closed traverse.
Line segments that limit the polygon are called sides points to cut the sides are the vertices of the polygon, each of these vertices are designated by a letter.
The perimeter of a polygon is the sum of the lengths of all sides.

ESTATES CLASSIFIED BY THE NUMBER OF SIDES:

· Triangle: 3 sides.
· Quadrilaterals: 4 sides.
· Pentagons: 5 sides.
· Hexagons: 6 sides.
· Heptagon, 7 sides.
· Octagon, 8 sides.
· enneagon: 9 sides.
· Decagon, 10 sides.
· hendecagon: 11 sides.
· dodecagons: 12 sides.
· Pentadecágonos: 15 sides.
· irregular icosagon: 20 sides.

polygons with number of sides other than those listed above are referred to as polygons ... (Place the number of sides has the polygon) ... sides. Example: polygon of 17 sides.

EQUALITY OR INEQUALITY IN THEIR SIDES AND ANGLES:

· Regular Polygon: is the one with all sides and angles equal. To be regulated should comply with two conditions.

regular polygons can be convex or convex, the latter are also called star polygon.

· Irregular polygon: it is one that has no equal among themselves, or their sides or angles.

· Polygon Registered: is one that has all its vertices in a mass circle.

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· Polygon circumscribed: it is one that has all its sides tangent to the same circle.