1 Define in full detail what it means for a relation R to be

(1) Define, in full detail, what it means for a relation R to be an equivalence relation on a set S. (2) Is the relation R on P(S) defined by ARB if A intersection B = an equivalence relation?

Solution

An equivalence relation on a set S is a subset of SxS, i.e., a collection  of ordered pairs of elements of S, satisfying certain properties. Write \"\" to mean  is an element of , and we say \" is related to ,\" then the properties are

1. Reflexive:  for all ,

2. Symmetric:  implies  for all

3. Transitive:  and  imply  for all ,

When all these properties satisfy then it is an equivalence relation.

b) Given,

A ( Intersection ) B is an equivalence relation.

Lets check....

1) A ( Intersection ) A = A for all A...........satisfied reflexive property

2) A ( Intersection ) B ==> B ( Intersection ) A ......true.................satisfied...symmetric property

3) A ( Intersection ) B and B ( Intersection ) C

that means

( A ( Intersection ) B ) intersection ( B ( Intersection ) C ) doesnot imply A ( Intersection ) C.......................doesnot satisfy transitive property

Hence it is not an equivalence relation...

NO