Show that the relation R in the set A of all the books in a library of a college, given by \(\mathrm{R}=\{(x, y): x\) and \(y\) have same number of pages \(\}\) is an equivalence relation.
Expert Answer
Solution:
\(A=\) all books in library.
\(R=\{(x, y): x\) and \(y\) have same number of pages \(\}\)
(a) Reflexive : \(R=\{(a, a): a\) and \(a\) have same number of pages \(\}\). True so reflexive.
(b) Symmetric: If \(R=\left\{\left(a_1, a_2\right): a_1\right.\) and \(a_2\) have same no. of pages \(\}\)
Thus, \(R=\left\{\left(a_2, a_1\right): a_2\right.\) and \(a_1\) will definitely have same no. of pages \(\}\). So, symmetric.
(c) Transitive : If \(R=\left\{\left(a_1, a_2\right)\right\}\) and \(R=\left\{\left(a_2, a_3\right)\right\}\)
So, \(a_1, a_2\) and \(a_3\) all three books will have same no. of pages. Thus, \(R=\left\{\left(a_1, a_3\right)\right\}\) is true. So reflexive. Therefore, equivalence relation.
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