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Prove that the Digraph of a partial order has no cycle of length greater than 1.
Assume that there exists a cycle of length n ≥ 2 in the digraph of a partial order ≤ on a set A. This entails that there are n distinct elements a1 , a2 , a3 , ..., an like that a1 ≤ a2 , a2 ≤ a3 , ..., an-1 ≤ an and an ≤ a1 . Applying the transitivity n-1 times on a1 ≤ a2 , a2 ≤ a3 , ..., an-1 ≤ an , we get a1 ≤ an .As relation ≤ is anti-symmetric a1 ≤ an and an ≤ a1 together entails that a1 = an . This is contrary to the fact that all a1, a2, a3... an are distinct. So, our assumption that there is a cycle of length n ≥ 2 in the digraph of a partial order relation is wrong.
Rick is order a latest triangular sail for his boat. He needs knowing the area of the sail. Which formula will he use? The area of a triangle is 1/2 times the length of the bas
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how i do project in linear programming in agriculture
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i don''t understand what my teacher when she talks about when she talks about cosecutive integers etc... so can u help me???
statement of gauss thm
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