Mathematics Exam > Mathematics Questions > f = (1 2 3) (1 2) isa)odd permutationb)even p...
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f = (1 2 3) (1 2) is
- a)odd permutation
- b)even permutation
- c)Both (a) and (b)
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
f = (1 2 3) (1 2) isa)odd permutationb)even permutationc)Both (a) and ...
we have (1 2 3) (1 2) = (1 2) (1 3) (1 2) which is a product of odd number of transpositions.
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f = (1 2 3) (1 2) isa)odd permutationb)even permutationc)Both (a) and ...
Explanation:
To determine whether the permutation f = (1 2 3) (1 2) is an odd permutation or an even permutation, we need to understand the concept of parity in permutations.
Parity of a Permutation:
In mathematics, the parity of a permutation refers to whether the permutation can be expressed as an even number of transpositions (even permutation) or an odd number of transpositions (odd permutation).
Transposition:
A transposition is a permutation that swaps two elements and leaves all other elements unchanged. For example, (1 2) is a transposition that swaps the positions of 1 and 2.
Determining the Parity:
To determine the parity of a permutation, we count the number of transpositions required to obtain the given permutation.
In the given permutation f = (1 2 3) (1 2), we have two transpositions:
- The first transposition is (1 2 3), which swaps the positions of 1 and 2, and leaves 3 unchanged.
- The second transposition is (1 2), which swaps the positions of 1 and 2, and leaves 3 unchanged.
Therefore, the permutation f can be expressed as the composition of two transpositions.
Odd Permutation:
An odd permutation is a permutation that requires an odd number of transpositions to obtain the given permutation.
Even Permutation:
An even permutation is a permutation that requires an even number of transpositions to obtain the given permutation.
Answer:
In this case, the permutation f = (1 2 3) (1 2) can be expressed as the composition of two transpositions, which is an even number of transpositions. Therefore, the permutation f is an even permutation.
To determine whether the permutation f = (1 2 3) (1 2) is an odd permutation or an even permutation, we need to understand the concept of parity in permutations.
Parity of a Permutation:
In mathematics, the parity of a permutation refers to whether the permutation can be expressed as an even number of transpositions (even permutation) or an odd number of transpositions (odd permutation).
Transposition:
A transposition is a permutation that swaps two elements and leaves all other elements unchanged. For example, (1 2) is a transposition that swaps the positions of 1 and 2.
Determining the Parity:
To determine the parity of a permutation, we count the number of transpositions required to obtain the given permutation.
In the given permutation f = (1 2 3) (1 2), we have two transpositions:
- The first transposition is (1 2 3), which swaps the positions of 1 and 2, and leaves 3 unchanged.
- The second transposition is (1 2), which swaps the positions of 1 and 2, and leaves 3 unchanged.
Therefore, the permutation f can be expressed as the composition of two transpositions.
Odd Permutation:
An odd permutation is a permutation that requires an odd number of transpositions to obtain the given permutation.
Even Permutation:
An even permutation is a permutation that requires an even number of transpositions to obtain the given permutation.
Answer:
In this case, the permutation f = (1 2 3) (1 2) can be expressed as the composition of two transpositions, which is an even number of transpositions. Therefore, the permutation f is an even permutation.
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