Assignment 4

Deadline: April 26 (Thursday Class), April 27(Friday Class)

Harshit Kumar

April 20, 2012

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 10 1 11 1 1

2.

1 0 1 00 1 0 11 0 1 00 1 0 1

3.

1 1 1 01 1 1 01 1 1 00 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 dividedby 6.

Q5. Which of the following matrices represent partial order relation? Also, generate a Hasse diagram for those that are partial order relation.

1.

1 0 11 1 00 0 1

2.

1 0 00 1 01 0 1

3.

1 0 1 00 1 1 00 0 1 11 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