Advertisements
Advertisements
प्रश्न
If \[X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}\] \[X \subseteq Y .\]
Advertisements
उत्तर
Given:
\[X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}\]
To prove:
\[X \subseteq Y\]
\[\text{ Let }: \]
\[ x_n = 8^n - 7n - 1, n \in N\]
\[ \Rightarrow x_1 = 8 - 7 - 1 = 0\]
\[\text{ For any n } \geqslant 2, \text{ we have }: \]
\[ x_n = 8^n - 7n - 1 = (1 + 7 )^n - 7n - 1\]
\[ \Rightarrow x_n = ^{n}{}{C}_0 + ^{n}{}{C}_1 . 7 + ^{n}{}{C}_2 . 7^2 + ^{n}{}{C}_3 . 7^3 + . . . +^{n}{}{C}_n . 7^n - 7n - 1\]
\[ \Rightarrow x_n = 1 + 7n + ^{n}{}{C}_2 . 7^2 + ^{n}{}{C}_3 . 7^3 + . . . + 7^n - 7n - 1 [ \because ^{n}{}{C}_0 = 1 and^{n}{}{C}_1 = n]\]
\[ \Rightarrow x_n = 7^2 {^{n}{}{C}_2 +^{n}{}{C}_3 . 7 + ^{n}{}{C}_4 7^2 + . . . + ^{n}{}{C}_n . 7^{n - 2} }\]
\[ \Rightarrow x_n = 49{^{n}{}{C}_2 + ^{n}{}{C}_3 . 7 + ^{n}{}{C}_4 7^2 + . . . + ^{n}{}{C}_n . 7^{n - 2} }\]
\[\text{ Thus, x_n is some positive integral multiple of 49 for all } n \geqslant 2 . \]
\[X \text{ consists of all those positive integral multiples of 49 that are of the form } 49{ ^{n}{}{C}_2 +^{n}{}{C}_3 . 7 + ^{n}{}{C}_4 7^2 + . . . +^{n}{}{C}_n . 7^{n - 2} } \text{ along with zero } . \]
\[Y = {49(n - 1): n \text{ in } N} \text{ implies that it consists of all integral multiples of 49 along with zero } . \]
\[ \therefore X \subseteq Y\]
\[\]
APPEARS IN
संबंधित प्रश्न
What universal set (s) would you propose for the following:
The set of right triangles.
What universal set (s) would you propose for the following:
The set of isosceles triangles.
Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
\[\left( A \cup B \right)' = A' \cap B'\]
If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
\[\left( A \cap B \right)' = A'B' .\]
For any two sets A and B, prove that
A ∩ B ⊂ A
For any two sets A and B, prove that A ⊂ B ⇒ A ∩ B = A
For any two sets A and B, show that the following statements are equivalent:
(i) \[A \subset B\]
(ii) \[A \subset B\]=ϕ
(iii) \[A \cup B = B\]
(iv) \[A \cap B = A .\]
For three sets A, B and C, show that \[A \cap B = A \cap C\]
For three sets A, B and C, show that \[A \subset B \Rightarrow C - B \subset C - A\]
For any two sets, prove that:
\[A \cap \left( A \cup B \right) = A\]
For any two sets A and B, prove that: \[A \cap B = \phi \Rightarrow A \subseteq B'\]
If A and B are sets, then prove that \[A - B, A \cap B \text{ and } B - A\] are pair wise disjoint.
For any two sets of A and B, prove that:
\[A' \cup B = U \Rightarrow A \subset B\]
For any two sets of A and B, prove that:
\[B' \subset A' \Rightarrow A \subset B\]
Show that for any sets A and B, A ∪ (B – A) = (A ∪ B)
Each set X, contains 5 elements and each set Y, contains 2 elements and \[\cup^{20}_{r = 1} X_r = S = \cup^n_{r = 1} Y_r\] If each element of S belong to exactly 10 of the Xr's and to eactly 4 of Yr's, then find the value of n.
For any two sets A and B, prove that :
\[A' - B' = B - A\]
Let A and B be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( B - A \right)\]
In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
how many can speak English only.
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
B ∪ D
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
A ∪ B ∪ C
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
A ∪ B ∪ D
If X and Y are subsets of the universal set U, then show that X ∩ Y ⊂ X
If A and B are subsets of the universal set U, then show that A ⊂ A ∪ B
If A and B are subsets of the universal set U, then show that A ⊂ B ⇔ A ∪ B = B
A, B and C are subsets of Universal Set U. If A = {2, 4, 6, 8, 12, 20} B = {3, 6, 9, 12, 15}, C = {5, 10, 15, 20} and U is the set of all whole numbers, draw a Venn diagram showing the relation of U, A, B and C.
Let A, B and C be sets. Then show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B, 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers. Find the number of families which buy newspaper A only.
If A = {1, 3, 5, 7, 9, 11, 13, 15, 17} B = {2, 4, ..., 18} and N the set of natural numbers is the universal set, then A′ ∪ (A ∪ B) ∩ B′) is ______.
For all sets A and B, A – (A ∩ B) is equal to ______.
