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Tcecarenko [31]
3 years ago
7

Let A={Φ,{Φ},{Φ,{Φ}}} and B={1,2}. How many relations from A into B?

Mathematics
1 answer:
ahrayia [7]3 years ago
5 0

Answer: There are 64 relations from A to B.

Step-by-step explanation:

Number of relation from set A to B = 2^{\text{ (Number of eklemnts in A x B)}}

=2^{\text{(Number of elements in set A  ) x (Number of elements in set B)}}

Since, A={Φ,{Φ},{Φ,{Φ}}} and B={1,2}

i.e. Number of elements in set A = 3

Number of elements in set B = 2

So, Number of relation from set A to B = 2^{3\times 2}

= 2^6= 64

Hence, there are 64 relations from A to B.

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y′′ −y = 0, x0 = 0 Seek power series solutions of the given differential equation about the given point x 0; find the recurrence
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\displaystyle y''(x) = \sum_{n=2}^\infty n (n-1) a_nx^{n-2} = \sum_{n=0}^\infty (n+2) (n+1) a_{n+2} x^n

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Substitute these into the given differential equation:

\displaystyle \sum_{n=0}^\infty (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^\infty a_nx^n = 0

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\begin{cases}a_0 = y(0) \\ a_1 = y'(0) \\\\ a_{n+2} = \dfrac{a_n}{(n+2)(n+1)} & \text{for }n\ge0\end{cases}

Since the n-th coefficient depends on the (n - 2)-th coefficient, we split n into two cases.

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k=3 \implies n=6 \implies a_6 = \dfrac{a_4}{6\cdot5} = \dfrac{a_0}{6\cdot5\cdot4\cdot3\cdot2\cdot1}

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a_{n=2k+1} = \dfrac{a_1}{(2k+1)!}

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\boxed{\displaystyle y(x) = a_0 \sum_{k=0}^\infty \frac{x^{2k}}{(2k)!} + a_1 \sum_{k=0}^\infty \frac{x^{2k+1}}{(2k+1)!}}

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