Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that- A x (B ∩C) = (A x B) ∩(A x C)
- A x C is a subset of B x D
- A x (B ∩C) = (A x B) ∩(A x C)
- A x C is a subset of B x D
Solution and Explanation
(i) To verify: A x (B∩C) = (A x B)∩(A x C)
We have B ∩C = {1, 2, 3, 4}∩{5, 6} = Φ
∴L.H.S. = A x (B∩C) = A x Φ = Φ
A x B = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4)}
A x C = {(1, 5), (1, 6), (2, 5), (2, 6)}
∴ R.H.S. = (A x B) ∩(A x C) = Φ
∴L.H.S. = R.H.S
Hence, A x (B ∩C) = (A x B) ∩(A x C)
(ii) To verify: A x C is a subset of B x D
A x C = {(1, 5), (1, 6), (2, 5), (2, 6)}
B x D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}
We can observe that all the elements of set A x C are the elements of set B x D.
Therefore, A x C is a subset of B x D.
Top Questions on Relations and functions
- If the domain of the function \( f(x) = \log_e \left( \frac{2x + 3}{4x^2 + x - 3} \right) + \cos^{-1} \left( \frac{2x - 1}{x + 2} \right) \) is \( (\alpha, \beta] \), then the value of \( 5\beta - 4\alpha \) is equal to
- JEE Main - 2024
- Mathematics
- Relations and functions
- Consider the relations $R_1$ and $R_2$ defined as \[a R_1 b \iff a^2 + b^2 = 1 \quad \text{for all } a, b \in \mathbb{R},\]and \[(a, b) R_2 (c, d) \iff a + d = b + c \quad \text{for all } (a, b), (c, d) \in \mathbb{N} \times \mathbb{N}.\]Then:
- JEE Main - 2024
- Mathematics
- Relations and functions
- Let the relations \( R_1 \) and \( R_2 \) on the set
\( X = \{ 1, 2, 3, \dots, 20 \} \) be given by
\( R_1 = \{ (x, y) : 2x - 3y = 2 \} \) and
\( R_2 = \{ (x, y) : -5x + 4y = 0 \} \).
If \( M \) and \( N \) be the minimum number of elements required to be added in \( R_1 \) and \( R_2 \), respectively, in order to make the relations symmetric, then \( M + N \) equals:- JEE Main - 2024
- Mathematics
- Relations and functions
- The interval in which the function \( f(x) = x^x, \, x>0 \), is strictly increasing is:
- JEE Main - 2024
- Mathematics
- Relations and functions
- The function \( f(x) = \frac{x^2 + 2x - 15}{x^2 - 4x + 9} \), \( x \in \mathbb{R} \) is:
- JEE Main - 2024
- Mathematics
- Relations and functions
Questions Asked in CBSE Class XI exam
- Distinguish between the following pairs of sentences.
(i) He was visibly moved.
(ii) He was visually impaired.- CBSE Class XI
- The adventure
- How, according to you, can peace and liberty be maintained in a state?
- CBSE Class XI
- The tale of melon City
Figures 9.20(a) and (b) refer to the steady flow of a (non-viscous) liquid. Which of the two figures is incorrect ? Why ?
- CBSE Class XI
- Pressure
- Describe the change in hybridisation (if any) of the Al atom in the following reaction.
\(AlCl_3+Cl^- \rightarrow AlCl_4^-\)- CBSE Class XI
- Hybridisation
- Why did the narrator of the story want to forget the address?
- CBSE Class XI
- The address
Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
