Solution: Consider the fuzzy relation:

R = $ \left[ \begin{array} & 1& 0.8 & 0 & 0.1 & 0.2\\ 0.8&1&0.4&0&0.9\\ 0&0.4&1&0&0\\ 01&0&0&1&0.5\\ 0.2&0.9&0&0.5&1\\ \end{array} \right]$

Fuzzy tolerance relation is having the property of reflexivity, symmetry but not transitivity.

$M_R (x_1, x_2) = 0.8$ $M_R (x_2, x_3) = 0.4$

From the relation R, we have,

$M_R (x_1, x_3) = 0$ - - - - (1)

But, on calculating we obtain,

$M_R (x_1, x_3) = min [ M_R (x_1, x_2), M_R (x_2, x_3)]$

= min (0.8, 0.4)

= 0.4 - - - - (2)

Equation [1] and [2] are not equal.

So the given relation is fuzzy tolerance relation.

Now assume $\lambda = 0.7$, then the relation becomes.

R = $\left[\begin{array} &1&1&0&0&0\\ 1&1&0&0&1\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&1&0&0&1 \end{array}\right] $

Now, $x_1, x_2) \in R$ Now,

$(x_2, x_5) \in R$

But $(x_1, x_5) \notin R$

Hence R 0.7 is crisp tolerance relation.