Advertisements
Advertisements
प्रश्न
If A × B ⊆ C × D and A × B ≠ ϕ, prove that A ⊆ C and B ⊆ D.
उत्तर
\[Let: \]
\[(x, y) \in (A \times B)\]
\[ \therefore x \in A, y \in B\]
\[Now, \]
\[ \because (A \times B) \subseteq (C \times D) \]
\[ \therefore (x, y) \in (C \times D) \]
\[Or, \]
\[x \in C \text{ and }y \in D\]
\[\text{ Thus, we have:} \]
\[ A \subseteq C B \subseteq D\]
APPEARS IN
संबंधित प्रश्न
If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
State whether the following statement is true or false. If the statement is false, rewrite the given statement correctly.
If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}.
State whether the following statement is true or false. If the statement is false, rewrite the given statement correctly.
If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ.
If A = {–1, 1}, find A × A × A.
Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that A × (B ∩ C) = (A × B) ∩ (A × C)
Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that A × C is a subset of B × D
Let A and B be two sets such that n(A) = 3 and n (B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.
The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0, 1). Find the set A and the remaining elements of A × A.
If A = {1, 2, 3} and B = {2, 4}, what are A × B, B × A, A × A, B × B and (A × B) ∩ (B × A)?
If A and B are two set having 3 elements in common. If n(A) = 5, n(B) = 4, find n(A × B) and n[(A × B) ∩ (B × A)].
If A = {1, 2}, from the set A × A × A.
If A = {2, 3}, B = {4, 5}, C ={5, 6}, find A × (B ∪ C), A × (B ∩ C), (A × B) ∪ (A × C).
If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find
(iii) A × (B ∪ C)
Prove that:
(i) (A ∪ B) × C = (A × C) ∪ (B × C)
(ii) (A ∩ B) × C = (A × C) ∩ (B×C)
Find the domain of the real valued function of real variable:
(v) \[f\left( x \right) = \frac{x^2 + 2x + 1}{x^2 - 8x + 12}\]
Find the domain of the real valued function of real variable:
(ii) \[f\left( x \right) = \frac{1}{\sqrt{x^2 - 1}}\]
Find the domain of the real valued function of real variable:
(iv) \[f\left( x \right) = \frac{\sqrt{x - 2}}{3 - x}\]
Find the domain and range of the real valued function:
(ii) \[f\left( x \right) = \frac{ax - b}{cx - d}\]
Find the domain and range of the real valued function:
(iii) \[f\left( x \right) = \sqrt{x - 1}\]
Find the domain and range of the real valued function:
(iv) \[f\left( x \right) = \sqrt{x - 3}\]
Find the domain and range of the real valued function:
(v) \[f\left( x \right) = \frac{x - 2}{2 - x}\]
Find the domain and range of the real valued function:
(vii) \[f\left( x \right) = - \left| x \right|\]
Find the domain and range of the real valued function:
(ix) \[f\left( x \right) = \frac{1}{\sqrt{16 - x^2}}\]
Find the domain and range of the real valued function:
(x) \[f\left( x \right) = \sqrt{x^2 - 16}\]
If f(x) be defined on [−2, 2] and is given by \[f\left( x \right) = \begin{cases}- 1, & - 2 \leq x \leq 0 \\ x - 1, & 0 < x \leq 2\end{cases}\] and g(x)
\[= f\left( \left| x \right| \right) + \left| f\left( x \right) \right|\] , find g(x).
Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine A × B
Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine is A × B = B × A?
Let A = {–1, 2, 3} and B = {1, 3}. Determine B × A
Let A = {–1, 2, 3} and B = {1, 3}. Determine B × B
A = {x : x ∈ W, x < 2} B = {x : x ∈ N, 1 < x < 5} C = {3, 5} find A × (B ∩ C)
State True or False for the following statement.
If A × B = {(a, x), (a, y), (b, x), (b, y)}, then A = {a, b}, B = {x, y}
