Advertisements
Advertisements
Question
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Advertisements
Solution
f : A × B → B × A is defined as f(a, b) = (b, a).
Let (a1, b1), (a2, b2) ∈ A × B such that f(a1, b1) = f(a2, b2).
⇒ (b1, a1) = (b2, a2)
⇒ b1 = b2 and a1 = a2
⇒ (a1, b1) = (a2, b2)
∴ f is one-one.
Now, let (b, a) ∈ B × A be any element.
Then, there exists (a, b) ∈ A × B such that f(a, b) = (b, a). ...[By definition of f]
∴ f is onto.
Hence, f is bijective.
APPEARS IN
RELATED QUESTIONS
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 2), (b, 1), (c, 1)}
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = ex g(x) = loge x .
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
If f(x) = |x|, prove that fof = f.
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−1
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
If f : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
If f : R → R is defined by f(x) = x2, write f−1 (25)
If f : R → R is given by f(x) = x3, write f−1 (1).
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
The function
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Mark the correct alternative in the following question:
Let f : R→ R be defined as, f(x) = \[\begin{cases}2x, if x > 3 \\ x^2 , if 1 < x \leq 3 \\ 3x, if x \leq 1\end{cases}\]
Then, find f( \[-\]1) + f(2) + f(4)
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
f(x) = `x/2`
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
g(x) = |x|
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
The function f: R → R defined as f(x) = x3 is:
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
