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VinnyCee
VinnyCee is offline
#3
Feb21-07, 03:15 AM
P: 492
Let [itex]P(n)[/itex] be [tex]\left(A_1\,\cap\,A_2\,\cap\,\dots\,\cap\,A_n\right)\,\cup\,B\,=\,\left( A_1\,\cup\,B\left)\,\cap\,\left(A_2\,\cup\,B\right)\,\cap\,\dots\,\cap\ ,\left(A_n\,\cup\,B\right)[/tex]

Then [itex]\forall\,n\,\left(P(n)\right)[/itex], right?



First I do the basis step for [itex]P(1)[/itex]:

[tex]A_1\,\cup\,B\,=\,\left(A_1\,\cup\,B\right)[/tex]



Now I need to show that [itex]P(k)\,\longrightarrow\,P(k\,+\,1)[/itex]?

For [itex]P(k)[/itex] (Also, assume it is true for induction):

[tex]\left(A_1\,\cap\,A_2\,\cap\,\dots\,\cap\,A_k\right )\,\cup\,B\,=\,\left(A_1\,\cup\,B\left)\,\cap\,\left(A_2\,\cup\,B\right )\,\cap\,\dots\,\cap\,\left(A _k\,\cup\,B\right)[/tex]


EDIT: Removed errors