Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Let \( S_n \) denote the set of all algebraic permutations of \( n \) objects. (a) How many elements are in the set \( S_n \)? [Input box for answer] Let \(\sigma = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 5 & 1 & 3 \end{bmatrix}\) and let \(\tau = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 1 & 2 & 4 \end{bmatrix}\) be algebraic permutations in \( S_5 \). (b) Find \(\sigma(1)\). [Input box for answer] (c) Find \(\tau \circ \sigma(1)\). [Input box for answer] (d) Find \(\sigma(2)\). [Input box for answer] (e) Find \(\tau \circ \sigma(2)\). [Input box for answer] (f) Find \(\sigma(3)\). [Input box for answer] (g) Find \(\tau \circ \sigma(3)\). [Input box for answer] (h) Find \(\tau \circ \sigma: \{1,2,3,4,5\} \to \{1,2,3,4,5\}\). Options: - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 5 & 4 & 2 \end{bmatrix}\) - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}\) [Correct] - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 5 & 3 & 1 & 2 \end{bmatrix}\) - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 5 & 4 & 3 \end{bmatrix}\)
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Transcribed Image Text:Let \( S_n \) denote the set of all algebraic permutations of \( n \) objects. (a) How many elements are in the set \( S_n \)? [Input box for answer] Let \(\sigma = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 5 & 1 & 3 \end{bmatrix}\) and let \(\tau = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 3 & 1 & 2 & 4 \end{bmatrix}\) be algebraic permutations in \( S_5 \). (b) Find \(\sigma(1)\). [Input box for answer] (c) Find \(\tau \circ \sigma(1)\). [Input box for answer] (d) Find \(\sigma(2)\). [Input box for answer] (e) Find \(\tau \circ \sigma(2)\). [Input box for answer] (f) Find \(\sigma(3)\). [Input box for answer] (g) Find \(\tau \circ \sigma(3)\). [Input box for answer] (h) Find \(\tau \circ \sigma: \{1,2,3,4,5\} \to \{1,2,3,4,5\}\). Options: - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 3 & 5 & 4 & 2 \end{bmatrix}\) - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 4 & 5 & 1 \end{bmatrix}\) [Correct] - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 5 & 3 & 1 & 2 \end{bmatrix}\) - \(\begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 5 & 4 & 3 \end{bmatrix}\)
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