Consider the recursively defined set S: Basis Step: The unit circle is in S. Recursive Step: if x is in S, then x with a line th
rough any diameter is in S. (a) (4 points) Prove that: is in S. (b) ( 6 points) For an element x ∈ S, define V (x) be the number of vertices (i.e. the number of intersections of lines and arcs and lines with lines), let E(x) to be the number of edges (line segments or arcs between vertices), and let F(x) be the number of faces. Prove that for any x ∈ S that F + V = E + 1. (Please use structural induction.)
1 answer:
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Answer:
21) falls
22) vertical
23) rises
24) rises
25) falls
26) horizontal
27) rises
28) horizontal
29) falls
Step-by-step explanation:
We shall use the slope formula to calculate the slopes of the line passing through each pair of point.
21: (3,1), (2,6)
The slope is
.
A negative slope means this line falls
22) (-2,5), (-2,4)
The slope is = undefined
An undefined slope means this line is vertical
23) The given point is (0,8) (2,10)
The slope is
A positive slope means this line rises
24) The points are (-3,-3), (3,1)
The slope is
A positive slope means this line rises
25) The points are: (5,0) (6,-2)
Slope:
A negative slope means this line falls
26) The points are (-2,-8) , (5,-8).
Slope:
A zero slope means this line is horizontal
27) The points are: (-1,2) , (5,3)
Slope:
A positive slope means this line rises
28) The points are: , (-1,4)
Slope:
A zero slope means the line is horizontal
29) The points are:
Slope:
A negative slope means this line falls.