This document provides the details of Assignment 4 which is due on April 26th or 27th depending on the class. It lists 7 questions related to concepts like transitive closure, equivalence relations, partitions, partial order relations, and Hasse diagrams. Students are asked to find transitive closures of relations, identify which relations are equivalence relations, determine if matrices represent equivalence relations, identify which subsets are partitions, determine if matrices represent partial order relations and generate Hasse diagrams, determine if a given diagram represents a partial order, and draw the Hasse diagram for a given set inclusion diagram.

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