Thread: Equivalence Relations and Partial Orderings

1. Give an interpretation of the equivalence classes for the equivalence relation R in: Let R be the relation on the set of ordered pairs of positive integers such that ((a,b),(c,d)) ℇ R if and only if ad = bc.

2) suppose that R1 and R2 are equivalence relations on the set S. Determine whether each of these combinations of R1 and R2 must be an equivalence relation.
a) R1 U R2
b) R1 ∩ R2
c) R1 ⊕ R2

3) Do we necessarily get an equivalence relation when we form the symmetric closure of the reflexive closure of the transitive closure of a relation?

4) Give an example of an infinite lattice with both a least and a greatest element.

Thank You

6) Answer these questions for the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, |}. Are the answers correct?

(a) Find the maximal elements.
27, 48, 60, 72

(b) Find the minimal elements.
2, 9

(c) Is there a greatest element?
No.

(d) Is there a least element?
No.

(e) Find all upper bounds of {2, 9}.
18, 36, 72

(f) Find the least upper bound of {2, 9}, if it exists.
18

(g) Find all lower bounds of {60, 72}.
2,4,6,12

(h) Find the greatest lower bound of {60, 72}, if it exists.
12

Can some teach me how to draw Hasse diagram! ---Got it!!! yeah