Question

Let   f:  A ⟶ B be a function. If  f is bijection then   f − 1 is a bijective function from B to A.

Group of answer choices

True

False

Answer 1

solution:

given

  since f : A$\rightarrow$B be bijective f has invers $^{^{f-1}}$ :B$\rightarrow$A

suppose $^{f-1}$ (b) = $^{f-1}$ ($_{b2}$) for some b₁ ,b₂ in B since f is bijective

f is injective and sub jective since the f is subjective there are elemments

x_{1 ,} x₂ in a such that b₁ = f (x₁)

b₂ = f (x₂)

since f ^-1 (b₁) = f ^-1 (b₂)

we have f ^-1 (f(x₁))= f^-1 (f(x₂)) holds by defination of invers function

f^-1 (f(x₁) = x₁ and f^-1(f(x₂)) = x₂

$\therefore$ hence x ₁ = x₂$\Rightarrow$ f(x₁) = f(x₂) $\Rightarrow$ y₁ = y₂

y₁ = y₂$\Rightarrow$ f ^-1 : B$\rightarrow$A is injective

note since f is a function ,x₁ = x₂$\Rightarrow$ f(x₁) = f(x₂)

since f is a function from A to B for any x$\epsilon$A there is an y$\epsilon$B

$\therefore$ y = f(x) $\Rightarrow$ f^-1 (y) = f ^-1(f(x))=x

hence for any x$\epsilon$A there is an y$\epsilon$B such that f ^-1 (y) = x

hence f ^-1 is a subjection f^-1 ia a bijection

the answer is true