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Consider a database whose universe is a finite set of vertices V and whose unique relation .E is binary and encodes the edges of an undirected (resp., directed) graph G: (V, E). Each undirected edge between the nodes o and u (resp., directed edge from the node v to the node u) is encoded by the two atoms E (v, u) and E (u, v) (resp., by the single atom E (v, u)).
Consider the pairs of stucture (undirected (resp., directed) graphs) shown in Fig. 1. Suppose that the graphs are encoded in a database as explained above. For each pair, answer the following questions:
1. What is the smallest quantifier rank k for which the spoiler wins the k-move Ehrenfeucht-Fraisse game on the pair of structure?
2. Derive a Boolean first-order query from your winning strategy that is true on one structure but not on the other (you can use the equality relation between vertices).
Method of disks or the method of rings One of the simple methods for getting the cross-sectional area is to cut the object perpendicular to the axis of rotation. Carrying out
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HE IGHTS AND DISTANCES If the angle of elevation of cloud from a point 'h' meters above a lake is α and the angle of depression of its reflection in the lake is β , prove
Evaluate each of the following. (a) 25 1/2 (b) 32 1/5 Solution (a) 25 1/2 Thus, here is what we are asking in this problem. 2
Example of Implicit differentiation So, now it's time to do our first problem where implicit differentiation is required, unlike the first example where we could actually avoid
y 2 = t 2 - 3 is the actual implicit solution to y'= t/y, y(2) = -1. At such point I will ask that you trust me that it is actually a solution to the differential equation. You w
Identify the surface for each of the subsequent equations. (a) r = 5 (b) r 2 + z 2 = 100 (c) z = r Solution (a) In two dimensions we are familiar with that this
a) Let V = f1, 2, :::, 7g and define R on V by xRy iff x - y is a multiple of 3. You should know by now that R is an equivalence relation on V . Suppose that this is so. Explain t
Forecasting By Using Least Squares Data have been kept of sales over the last seven years Year 1 2 3 4 5 6
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