Suppose $$A$$ is a finite set with $$n$$ elements. The number of elements in the Largest equivalence relation of $$A$$ is

The binary relation R = {(1, 1)}, (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4) } on the set A = { 1, 2, 3, 4} is

In a room containing 28 people, there are 18 people who speak English, 15 people who speak Hindi and 22 people who speak Kannada, 9 persons speak both English and Hindi, 11 persons speak both Hindi and Kannada where as 13 persons speak both Kannada and English. How many people speak all three languages?

(a) Find the points of local maxima and minima, if any, of the following function defined $$0 \le x \le 6.\,\,\,{x^3} - 6{x^2} + 9x + 15$$

(b) Integrate $$\,\,\,\int\limits_{ - \pi }^\pi {x\,\cos \,x\,dx} $$