GRO 5. Consider the infinite set of positive integers {1, 2, 3,..}. Find a way to name definite subsets as lines and other definite subsets as planes in such a manner that the incidence axioms will automatically hold as in the manner of the previous tetra- hedron model. (Hint: Let one line consist of the points 3, 4, 5,.. .) ing model in which there is fall

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Chapter2: Second-order Linear Odes
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Please answer question 5 (2.3)

Which axiom is not satisfied?
3. Consider the following system of points, lines, and planes:
Points: S {1, 2, 3.4, 5}
Lines: (1, 2. 3}, {1.4). {1, 5}. {2, 4}. {2, 5}. {3, 4}, {3, 5}. {4. 5}
Planes: {1, 2, 3, 4}. {1, 2, 3, 5}, {1, 2, 3}. {1, 4, 5}. {2, 4, 5}, {3, 4, 5}
Which incidence axioms are satisfied here?
4. Consider the following system of points, lines, and planes:
{1, 2, 3, 4}
Lines: {1, 2, 4}. {1, 3}, {2, 3}, {3, 4}
Planes: {1, 2, 3}, {1, 3, 4}, {2, 3, 4}
Points: S =
ory
Which incidence axioms are satisfied in this model?
GROUP B
5. Consider the infinite set of positive integers {1, 2, 3, . . .}. Find a way to name
definite subsets as lines and other definite subsets as planes in such a manner that
the incidence axioms will automatically hold as in the manner of the previous tetra-
hedron model. (Hint: Let one line consist of the points 3, 4, 5, . . .)
6. The 7-Point Projective Plane Consider the following model, in which there is
only one plane, the universal set S itself.
line passing
through 2, 4, 7
{1, 2, 3, 4, 5, 6, 7}
{1, 2, 3}, l2 = {1, 4, 5}, lz = {3,7, 5}, €4 = {1, 6, 7},
ls = {2, 5, 6}, lo = {3, 4, 6}, l7 = {2, 4, 7}
{1, 2, 3, 4, 5, 6, 7}
Points: S
%3D
Lines: l1
%3D
%3D
4
6.
%3D
Plane: P
%3D
Which incidence axioms are satisfied? (Compare this model with the committee
structure in the Flying Aviators Club of Problem 11, Section 2.2.)
7. The Four-Point Affine Plane Show that the model depicted here for four points
satisfies the axioms for an affine plane, consisting of Axiom I-1, the pertinent part
of Axiom I-5 dealing with a single plane, and the Parallel Postulate (a unique line
exists that passes through a given point parallel to-does not intersect-a given line).
3
Transcribed Image Text:Which axiom is not satisfied? 3. Consider the following system of points, lines, and planes: Points: S {1, 2, 3.4, 5} Lines: (1, 2. 3}, {1.4). {1, 5}. {2, 4}. {2, 5}. {3, 4}, {3, 5}. {4. 5} Planes: {1, 2, 3, 4}. {1, 2, 3, 5}, {1, 2, 3}. {1, 4, 5}. {2, 4, 5}, {3, 4, 5} Which incidence axioms are satisfied here? 4. Consider the following system of points, lines, and planes: {1, 2, 3, 4} Lines: {1, 2, 4}. {1, 3}, {2, 3}, {3, 4} Planes: {1, 2, 3}, {1, 3, 4}, {2, 3, 4} Points: S = ory Which incidence axioms are satisfied in this model? GROUP B 5. Consider the infinite set of positive integers {1, 2, 3, . . .}. Find a way to name definite subsets as lines and other definite subsets as planes in such a manner that the incidence axioms will automatically hold as in the manner of the previous tetra- hedron model. (Hint: Let one line consist of the points 3, 4, 5, . . .) 6. The 7-Point Projective Plane Consider the following model, in which there is only one plane, the universal set S itself. line passing through 2, 4, 7 {1, 2, 3, 4, 5, 6, 7} {1, 2, 3}, l2 = {1, 4, 5}, lz = {3,7, 5}, €4 = {1, 6, 7}, ls = {2, 5, 6}, lo = {3, 4, 6}, l7 = {2, 4, 7} {1, 2, 3, 4, 5, 6, 7} Points: S %3D Lines: l1 %3D %3D 4 6. %3D Plane: P %3D Which incidence axioms are satisfied? (Compare this model with the committee structure in the Flying Aviators Club of Problem 11, Section 2.2.) 7. The Four-Point Affine Plane Show that the model depicted here for four points satisfies the axioms for an affine plane, consisting of Axiom I-1, the pertinent part of Axiom I-5 dealing with a single plane, and the Parallel Postulate (a unique line exists that passes through a given point parallel to-does not intersect-a given line). 3
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