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Prove that a simple graph is connected if and only if it has a spanning tree.
Ans: First assume that a simple graph G has a spanning tree T. T consists of every node of G. By the definition of a tree, there is a path among any two nodes of T. As T is a subgraph of G, there is a path among each pair of nodes in G. Hence G is connected.
Here now let G is connected. If G is a tree then nothing to prove. If G is not a tree, it must consist of a simple circuit. Let G has n nodes. We can choose (n - 1) arcs from G in such type of a way that they not form a circuit. It results into a subgraph comprising all nodes and only (n - 1) arcs. So by definition this subgraph is a spanning tree.
Determine if the acceleration of an object is given by a → = i → + 2 j → + 6tk → find out the object's velocity and position functions here given that the initial velocity is v
State the following statement as a disjunction (in DNF) as well using quantifiers: There does not exit a woman who has taken a flight on each airline in the world.
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Consider the following linear equations. x1-3x2+x3+x4-x5=8 -2x1+6x2+x3-2x4-4x5=-1 3x1-9x2+8x3+4x4-13x5=49
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Q. Converting Mixed Numbers to Improper Fractions? Ans. Converting a mixed number to an improper fraction is easy. A single multiplication, and then a single addition:
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