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Exercise:
forall sigma tau in mathcalS_n we have textsgnsigma tautextsgnsigma textsgntau.

Solution:
Proof. If tau_...tau_n can be obtained from ...n by a sequence of m transpositions then sigmatau ...sigmatau n can be obtained from sigma ...sigma n by a sequence of m transpositions simply apply sigma to each tuple in the process ...nrightarrow...rightarrow tau_...tau_nLongrightarrow sigma_...sigma_nrightarrow ...rightarrow sigmatau ...sigmatau n. If sigma ...sigma n can be obtained from ...n by p transpositions then sigmatau ...sigmatau n can be obtained from ...n by m+p transpositions ...nrightarrow ...rightarrow sigma ...sigma nrightarrow ... rightarrow sigmatau ... sigmatau n. textsgnsigmatau-- -textsgnsigma textsgntau
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Exercise:
forall sigma tau in mathcalS_n we have textsgnsigma tautextsgnsigma textsgntau.

Solution:
Proof. If tau_...tau_n can be obtained from ...n by a sequence of m transpositions then sigmatau ...sigmatau n can be obtained from sigma ...sigma n by a sequence of m transpositions simply apply sigma to each tuple in the process ...nrightarrow...rightarrow tau_...tau_nLongrightarrow sigma_...sigma_nrightarrow ...rightarrow sigmatau ...sigmatau n. If sigma ...sigma n can be obtained from ...n by p transpositions then sigmatau ...sigmatau n can be obtained from ...n by m+p transpositions ...nrightarrow ...rightarrow sigma ...sigma nrightarrow ... rightarrow sigmatau ... sigmatau n. textsgnsigmatau-- -textsgnsigma textsgntau
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eth, fs23, lineare algebra, proof
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(3, default)
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ENG (English)
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