Let A and B be sets. Show that f: \(A \times B → B \times A\) such that (a,b)=(b,a) is bijective function
Solution and Explanation
f: \(A \times B → B \times A\) is defined as f(a, b) = (b, a).
Let \((a_1,b_1)\), \((a_2,b_2)\) \(∈ A\times B\) such that \(f(a_1,b_1)=f(a_2,b_2)\).
\(⇒ (b_1,a_1)=(b_2,a_2)\)
\(⇒ b_1=b_2 \text{ and } a_1=a_2\)
\(⇒ (a_1,b_1)=(a_2,b_2)\)
∴ 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.
Top Questions on Relations and Functions
- Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = 10 - x^2 \), then:
- CUET (UG) - 2024
- Mathematics
- Relations and Functions
- Relation \( R \) on the set \( A = \{1, 2, 3, \ldots, 13, 14\} \) defined as \( R = \{(x, y) : 3x - y = 0\} \) is:
- CUET (UG) - 2024
- Mathematics
- Relations and Functions
- Match List I with List II
LIST I LIST II A. Range of y=cosec-1x I. R-(-1, 1) B. Domain of sec-1x II. (0, π) C. Domain of sin-1x III. [-1, 1] D. Range of y=cot-1x IV. \([\frac{-π}{2},\frac{π}{2}]\)-{0} Choose the correct answer from the options given below:- CUET (UG) - 2023
- Mathematics
- Relations and Functions
- If f(x) - 27x3 and g(x) = \((x)^{\frac{1}{3}}\), then gof(x) is :
- CUET (UG) - 2023
- Mathematics
- Relations and Functions
- The relation R in the set A = {1, 2, 3, 4} is given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)} is :
- CUET (UG) - 2023
- Mathematics
- Relations and Functions
Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
- For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?
- Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
- Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
Concepts Used:
Types of Functions
Types of Functions
One to One Function
A function is said to be one to one function when f: A → B is One to One if for each element of A there is a distinct element of B.
Many to One Function
A function which maps two or more elements of A to the same element of set B is said to be many to one function. Two or more elements of A have the same image in B.
Onto Function
If there exists a function for which every element of set B there is (are) pre-image(s) in set A, it is Onto Function.
One – One and Onto Function
A function, f is One – One and Onto or Bijective if the function f is both One to One and Onto function.
Read More: Types of Functions