sir i am not getting the concept of equivalence realtion even after watching ur video

Asked by  | 16th Apr, 2012, 12:54: PM

Expert Answer:

A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive

A relation R in a set A is called

  1. Reflexive, if (a, a) Î R, for every a Î A,
  2. Symmetric, if (a1, a2) Î R implies that (a2, a1) Î R, for all
    a1, a2 Î A.
  3. Transitive, if (a1, a2) Î R and (a2, a3) Î R implies that
    (a1, a3) Î R, or all a1, a2, a3 Î A.

Example:

1) Relation R in the set A of human beings in a town at a particular time given by R = {(xy): x is father of y}

(xx) ? R

As x cannot be the father of himself.

?R is not reflexive.

Now, let (xy) ?R.

x is the father of y.

y cannot be the father of y.

Indeed, y is the son or the daughter of y.

?(yx) ? R

? R is not symmetric.

Now, let (xy) ? R and (yz) ? R.

x is the father of y and y is the father of z.

x is not the father of z.

Indeed x is the grandfather of z.

? (xz) ? R

?R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive. Hence, it is not an equivalence relation.

2) Consider the relation R in the set A of all the books in a library of a college, given by R = {(xy): x and y have same number of pages}

Set A is the set of all books in the library of a college.

R = {xy): x and y have the same number of pages}

Now, R is reflexive since (xx) ? R as x and x has the same number of pages.

Let (xy) ? R ? x and y have the same number of pages.

y and x have the same number of pages.

? (yx) ? R

?R is symmetric.

Now, let (xy) ?R and (yz) ? R.

x and y and have the same number of pages and y and z have the same number of pages.

x and z have the same number of pages.

? (xz) ? R

?R is transitive.

Hence, R is an equivalence relation.

Answered by  | 8th May, 2012, 12:47: PM

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