Let F be the field with two elements {0,1} whose multiplication and addition hav
ID: 2261319 • Letter: L
Question
Let F be the field with two elements {0,1} whose multiplication and addition have the usual tables except that 1 + 1 = 0. Show that F2 is isomorphic to the smallest affine plane, described in Example 3 of the text. Show that P2(F) is isomorphic to the projective plane described in Example 6 of the text. This is the smallest projective plane; it has order 2 and is called the Fano plane.
EXAMPLE 3. Let the "points" be the four letters A, B, C, and D Let the "lines" be all six sets containing exactly two of these letters: [A, BI, {A,C), (A, D), (B, C), (B, D), and {C, D. Let "incidence" be set membership, as in Example 1. As an exercise, you can verify that this is a model for incidence geometry and that in this model the Euclidean parallel postulate does hold (see Figure 2.5). By Examples 1 and 3, the Euclidean parallel postulate is independent of the axioms of incidence geometryExplanation / Answer
Solution:
By the definition of "affine plane", we must verify the interpretations of the three incidence axioms and we must verify the Eucldean parallel property. If you've taken a course in analytic geometry, you know how to verify those. We sketch a few of the ideas: (1). To verify I-3, show that the points (0,0), (0,1), and (1,0) are not collinear by showing that any linear equation they all satisfy must have all three coefficients equal to 0.(2). To verify I-2, say coefficient [a eq 0] . Then (-c/a, 0) is one point on the line. Find another depending on whether b is 0 or not. (3). To verify I-1, let (u,v) and (s,t) be distinct points. Use your knowledge of analytic geometry to write a linear equation satisfied by those points. To show uniqueness, use Cramer's rule to find the unique solution to a pair of linearly independent linear equations. (4). To verify the Euclidean parallel property, first establish the result that two lines are parallel iff they have the same slope(handle the case of vertical lines separately). Then use the point-slope formula to determine the unique line parallel to a given line through a given point not on that line. Next we briefly describe the projective plane over F, denoted [P^{2}(F)] . Here both "points" and "lines" are equivalence classes of triples (x, y, z) of elements of F that are not all zero, where two such triples are considered equivalent if one is a nonzero constant multiple of the other. You can easily verify that this is an equivalence relation. Each such triple is referred to as homogeneous coordinates, and its equivalence class will be denoted [x, y, z]. We interpret incidence by the linear homogeneous equation ax+by+cz=0 when [x, y, z] is a "point" and [a, b, c] is a "line". We show that [P^{2}(F)] is isomorphic to the projective completion of [F^{2}] as follows: Map each "point" (x, y) of [F^{2}] to the "point" [x, y, 1] of [P^{2}(F)] . Map each "line" {(x,y)|ax+by+c=0} of [F^{2}] to the "line" [a, b, c] of [P^{2}(F)] . Verify easily that these mappings are one-to-one and preserve "incidence" for the affine plane. Next map the line at infinity in the projective completion to the "line" [0, 0, 1], i.e., to the "line" whose equation is z=0; it is the only "line" in [P^{2}(F)] that is not the image under our mapping of an affine line. A "point" on this line has homogeneous coordinates of the form [a, b, 0], where at least one of a,b is nonzero. We let this point correspond to the point at infinity common to all the lines parallel to the affine line
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