Zakat, as I understand it is for funding charity to the poor in the Ummah.
If one however lives in a state which has welfare funded through taxation, and one is a tax-payer, is it still neccessary to pay Zakat?
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Zakat can be viewed from two point of views: The one giving Zakat (i.e. you) and the one collecting Zakat (i.e. State).
From your point of view, it is an obligatory charity that you have to give to "purify" what Allah has given you.
From State point of view, it is the tax that it collects to run its affairs. If the State has taken it upon itself to do welfare work, then it also becomes an affair of the State.
In Quran, one of category to whom zakat can be given are the ones who are appointed to collect zakat, which essentially means the State.
Zakah expenditures are only for the poor and for the needy and for those employed to collect [zakah] and for bringing hearts together [for Islam] and for freeing captives [or slaves] and for those in debt and for the cause of Allah and for the [stranded] traveler - an obligation [imposed] by Allah . And Allah is Knowing and Wise.
In this case, you can subtract the amount of funds you give in taxes from your total zakat. If all (or more) taxes are given, then you don't need to pay any additional zakat.
Please note that this is one way to view Zakat, which some people disagree with. It is best to know the reasoning of different point-of-views and choose one, that your heart finds most peace in.