0

Consider $A$ = {$w, x, y, z$}. Determine:

(a) the number of possible relations on A, i.e., subsets of A×A

(b) the number of relations on A that are reflexive and symmetric.

(c) the number of relations on $A$ that are antisymmetric (consider the number of choices for a pair of element, say $(w, x)$ and $(x, w)$.

My Solutions: I was only able to solve the first part. Don't know how to solve the second or third part. (Answer (a) = $2^{16}$)

leonbloy
  • 66,202
George J. Adams
  • 883
  • You may see these links-

    for part (b): http://math.stackexchange.com/questions/12139/number-of-relations-that-are-both-symmetric-and-reflexive and for part (c) : http://math.stackexchange.com/questions/503979/how-to-find-the-number-of-anti-symmetric-relations

     – usermath Dec 18 '14 at 05:58

1 Answers1

2

Hint:

Think every relation as a binary square matrix of size 4 whose rows and columns are marked with characters $a,b,c,d$. If $a$-th row and $c$-th column is marked 1 then $(a,b) \in R$ otherwise not. Try to observe the properties of such a matrix if $R$ is reflexive, symmetric or antisymmetric .

arindam mitra
  • 1,481