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16 hrs of Maatr Movie English Subtitle Download. Online at Cinetorrent. Fans of Indian cinema from around the world.Q:
A directed graph $\mathbb{G}$ is a tree iff for every node $u$ there exists exactly one directed path from $u$ to any other node in $\mathbb{G}$
I do not have a clue on how to start this proof:
“A directed graph $\mathbb{G}$ is a tree iff for every node $u$ there exists exactly one directed path from $u$ to any other node in $\mathbb{G}$”
I tried to prove the if part, and I can prove the only if part is proved in the following way.
“$(\Rightarrow)$ Suppose that there exists a directed graph $\mathbb{G}$ that is not a tree. Then there exists two distinct nodes $u$ and $v$ of $\mathbb{G}$ such that there is a directed path from $u$ to $v$. Therefore, there are two distinct paths from $u$ to $v$, which contradicts the fact that the graph $\mathbb{G}$ is a tree.”
But I am not able to find any ideas about how to prove the if-part. Any help is appreciated.
A:
Here is one way to do it (which may not be the one you were trying). Suppose for the sake of contradiction that for every node there is more than one path. In particular, let $u$ be a node of the graph that does not have any path to the rest of the graph. Then there is a path starting at $u$ to some node $v$ (since the graph is connected). Then there is a path starting at $v$ back to $u$, which is a contradiction.
If you meant to prove that if $\mathbb{G}$ is a tree, then it is a directed graph, then one way to do it is to find the least path from $u$ to $v$. This is a path starting at $u$ and if there is no path from $u$ to $v$ then you have found the path in question. This is a contradiction, since otherwise you could get an additional path from $u
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How to use variables within a proc when database is empty?
proc sql;
select iif(v1=0,v2*0,v1);
quit;
With above code v1 is populated with previous value of v1.
proc sql;
select iif(v1=0,v2*0,v1);
quit;
Now with the above proc the above proc is executed regardless whether v1 is populated with previous value or not.
In my SQL code I am trying to test the value of v1 and need to use its result.
How should I do this?
A:
use a common block
?- setlib(lin, ‘plm’).
* /Users/harshj/SVN/plm/.libs/plm.old
?- create_program(plm, c).
* /Users/harshj/SVN/plm/lib/c++/plm.cpp:23:0:
plm(program,start,stop) :-
forall(e(+))(
+ conc(a, v1, a, v1, a),
d0c515b9f4
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