Question

a) What is the minimum number of pairs to form a non-zero reflexive relation on a set of n elements?

b) On the set R of real numbers, S is a relation defined as S = {(x,y)/X∈R, y ∈ R, x + y = xy}. Find a ∈ R such that ‘a’

is never the first element of an ordered pair in S. Also find b ∈ R such that ‘b’ is never the second element of an ordered pair in S.

c) Consider the function f(x)=\(\frac{3x + 4}{x-2}\),x≠2 Find a function on a suitable domain such that goƒ(x) = x = ƒog(x). 

Answer 1

a) n

b) a + y  = ay ⇒ ay - y = a ⇒ y (a - 1) = a

⇒ \(y = \frac{a}{a-1}\) ⇒ a ≠ 1

similarly; b ≠ 1 

c) go f(x) = x = f og (x) ⇒ g(x) = f^-1 (x)

f (x) = \(\frac{3x + 4}{x - 2}\) = y ⇒ 3x + 4 = y(x-2)

⇒ 3x + 4 = yx - 2y ⇒ yx - 3x = 2y + 4

⇒ x = \(\frac{2y + 4}{y - 3}\)

 ⇒ g(x) = \(\frac{2x + 4}{x-3}\)