3.2- Equivalence Relations 1) Indicate which of the following relations on the g

Question

3.2- Equivalence Relations

1) Indicate which of the following relations on the given sets are reflexive on a given set, which are symmetric, and which are transitive?

a) { (1,2) on {1,2)

b) less than or equal than natural numbers

c) = on N

d) < N

e) more than or equal to N

f) not equal to N

2. Let A = { 1, 2, 3}. List the ordered pairs and draw the diagraph of a relation on A with the given properties.

b) reflexive, not symmetric, and not transitive

c) not reflexive, symmetric, and not transitive

e) not reflexive, not symmetric ,and transitive

12) Suppose that R and S are equivalence relations on a set A. Prove that R? S is an equivalence relation on A

14) prove that if R is a symmetric, transitive relation o n A and the domain of R is A, then R is reflexive on A

3.3 Partitions

1. Describe four different partitions of the set of all students enrolled at a university

3 Describe the partition for each of the following equivalence relations

e. For ( x, y) and (u, v) E (element of) R X R, (x, y) S (u, v) iff xy = uv = 0 or xyuv > 0

8 List the ordered pairs in the equivalence relation on A = { 1, 2, 3, 4, 5} associated with these partitions

a) {{ 1, 2, }, { 3, 4, 5}}

b) {{ 1 }, {2}, {3,4}, {5}}

Explanation / Answer

1)

a).{(1,2)} on the set A = {1,2}. The relation is transitive.

(b)less than or equal than natural numbers The relation is re?exive and transitive

(c) = on N. The relation is re?exive, symmetric, and transitive.

(d) < on N. The relation is transitive.

(e) > on N. The relation is re?exive and transitive.

(f) Not equal to N . The relation is symmetric.

2) (b) re?exive, not symmetric, and not transitive.

{(1,1),(2,2),(3,3),(1,2),(2,3)}

(c) not re?exive, symmetric, and not transitive.

{(1,2),(2,1)}

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