Math  /  Discrete

QuestionIf A={1,2,3,4,5,6,7,8}A=\{1,2,3,4,5,6,7,8\}, then: (1,3,4)(5,1,2)(1,3,4) \circ(5,1,2) equals: (1,2,5,3,4)(1,2,5,3,4)
Select one: True False

Studdy Solution

STEP 1

1. We are given a set A={1,2,3,4,5,6,7,8}A = \{1, 2, 3, 4, 5, 6, 7, 8\} and two permutations (1,3,4)(1,3,4) and (5,1,2)(5,1,2).
2. The composition of permutations (1,3,4)(5,1,2)(1,3,4) \circ (5,1,2) needs to be calculated.
3. The resulting permutation should be compared to (1,2,5,3,4)(1,2,5,3,4) to determine if they are equal.
4. Permutation composition is performed right to left, meaning (1,3,4)(5,1,2)(1,3,4) \circ (5,1,2) is applied as (5,1,2)(5,1,2) first, then (1,3,4)(1,3,4).

STEP 2

1. Understand and apply the permutation (5,1,2)(5,1,2).
2. Apply the permutation (1,3,4)(1,3,4) to the result from Step 1.
3. Combine the results to find the final permutation.
4. Compare the final permutation to (1,2,5,3,4)(1,2,5,3,4) to determine if they are equal.

STEP 3

Apply the permutation (5,1,2)(5,1,2). This means: \begin{align*} 1 &\to 2 \\ 2 &\to 5 \\ 5 &\to 1 \\ \text{All other elements} &\to \text{themselves} \end{align*}

STEP 4

Now apply the permutation (1,3,4)(1,3,4) to the result of (5,1,2)(5,1,2): \begin{align*} 1 &\to 2 \to 2 \\ 2 &\to 5 \to 5 \\ 3 &\to 3 \to 4 \\ 4 &\to 4 \to 1 \\ 5 &\to 1 \to 3 \\ \text{All other elements} &\to \text{themselves} \end{align*} So, the intermediate results are: \begin{align*} 1 &\to 2 \\ 2 &\to 5 \\ 3 &\to 4 \\ 4 &\to 1 \\ 5 &\to 3 \\ 6 &\to 6 \\ 7 &\to 7 \\ 8 &\to 8 \end{align*}

STEP 5

Combine the results to form the final permutation: (1,2,5,3,4)(1,2,5,3,4)

STEP 6

Compare the final permutation to (1,2,5,3,4)(1,2,5,3,4): \begin{align*} 1 &\to 2 \\ 2 &\to 5 \\ 3 &\to 4 \\ 4 &\to 1 \\ 5 &\to 3 \end{align*} Since the final permutation matches (1,2,5,3,4)(1,2,5,3,4), the statement is TRUE.

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