Question

Let A = {a1, a2, a3, a4} where a1, > a2, > a3 > a4
The total number of unordered pairs of disjoint
subsets of A is equal to​


any person who have the dare this question!!!

Answer 1

Answer:

f : A → B, g : B → C are bijections ⇒ gof : A → C is a bijection Also g–1 : C → B and f–1 : B → A are bijections ⇒ f –1og–1 : C → A is a bijection. Let c be any element of C. Then ∃ an element b ∈ B such that g(b) = c ⇒ b = g–1(c) Also ∃ an element a ∈ A such that f(a) = b ⇒ a = f–1 (b) Now (gof) (a) = g(f(a)) = g(b) = c ⇒ a = (gof)–1 (c) ⇒ (gof)–1 (c) = a ——— (1) Also (f–1og–1) (c) = f–1(g–1(c)) =f–1(b) = a ——— (2) ∴ From (1) and (2); (gof)–1(c) = (f–1og–1) ( c) ⇒ (gof)–1 = f–1og–1.