Partial Order Set, Transitive Closure, Equivalence Relation,

1. Assignment 4
Deadline: April 26 (Thursday Class), April 27(Friday Class)
Harshit Kumar
April 20, 2012

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 not
represent 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.
Q7. Draw the Hasse diagram for the following set inclusion diagram. The original set S = {a, b, c, d}
1