Assignment 4Deadline: April 26 (Thursday Class), April 27(Friday Class)                       Harshit Kumar               ...
Q1. For the following sets, find the transitive closure.  1. {(a, c), (b, d), (c, a), (d, b), (e, d)}  2. {(b, c), (b, e), ...
Q7. Draw the Hasse diagram for the following set inclusion diagram. The original set S = {a, b, c, d}                     ...
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Assignment4

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Equivalence relation, Partial Order,

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Assignment4

  1. 1. Assignment 4Deadline: April 26 (Thursday Class), April 27(Friday Class) Harshit Kumar April 18, 2012
  2. 2. Q1. For the following sets, find the transitive closure. 1. {(a, c), (b, d), (c, a), (d, b), (e, d)} 2. {(b, c), (b, e), (c, e), (d, a), (e, b), (e, c)} 3. {(a, b), (a, c), (a, e), (b, a), (b, c), (c, a), (c, b), (d, a), (e, d)} 4. {(a, e), (b, a), (b, d), (c, d), (d, a), (d, c), (e, a), (e, b), (e, c), (e, e)} Q2. Assume that the Universe is a set of all people, then which of the following relations are equivalence relations? 1. {(p, q)|p and q were born on the same date}. 2. {(p, q)|p and q have same parents}. 3. {(p, q)|p and q share a common parent}. 4. {(p, q)|p and q are acquaintance}. 5. {(p, q)|p and q speak the same language} Q3. For the following matrices, Write your answer as YES if the matrix represents an equivalence relation. Write your answer as NO if the matrix does notrepresent an equivalence relation.   1 1 1 1. 0 1 1 1 1 1   1 0 1 0 0 1 0 1 2.  1 0 1 0  0 1 0 1   1 1 1 0 1 1 1 0 3.  1 1 1 0  0 0 0 1 Q4. For the subsets below, assume that the set of Integer Z is the universe; Which of the following subsets are partitions? 1. the set of integers divisible by 3, the set of integers leaving a remainder 2 when divided by 3, the set of integers leaving a remainder 1 when divided by 3. 2. the set of positive integers and the set of negative integers 3. the set of integers divisible by 2, the set of integers leaving a remainder of 1 when divided by 2. 4. the set of integers less than -200, the set of integers less than 200, and the set of integers greater than 200. 5. the set of integers leaving a remainder 0 when divided by 3, the set of integers divisible by 2, and the set of integers leaving a remainder 3 when divided by 6.Q5. Which of the following matrices represent partial order relation? Also, generate a Hasse diagram for those that are partial order relation.   1 0 1 1. 1 1 0 0 0 1   1 0 0 2. 0 1 0 1 0 1   1 0 1 0 0 1 1 0 3.  0 0 1 1  1 1 0 1 Q6. For the following diagram, determine if it represents a partial order or not? Explain your answer. 1
  3. 3. Q7. Draw the Hasse diagram for the following set inclusion diagram. The original set S = {a, b, c, d} 2

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