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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.
Realtors estimate that 23% of homes purchased in 2004 were considered investment properties. If a sample of 800 homes sold in 2004 is obtained what is the probability that at most
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Find out Least Common Multiple? The smallest number that is a common multiple of two numbers (that is, both numbers share the same multiple) is called the least common multiple
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There are 81 women teachers at Russell High. If 45% of the teachers in the school are women, how many teachers are there at Russell High? Use the proportion part/whole = %/100.
Using the example provided, Evaluate the area of the shaded region in terms of π. a. 264 - 18π b. 264 - 36π c. 264 - 12π d. 18π- 264 b. The area of the shaded r
24x+7=3x+10
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